diff --git "a/Astronomy/a_few_chapters_in_astronomy_1894.md" "b/Astronomy/a_few_chapters_in_astronomy_1894.md" new file mode 100644--- /dev/null +++ "b/Astronomy/a_few_chapters_in_astronomy_1894.md" @@ -0,0 +1,2996 @@ +Barcode: 8B 27A 052 +UC-NRLF + +A FEW CHAPTERS IN ASTRONOMY. + +Claudius Kennedy. + +YB 17013 + +LIBRARY +OF THE +UNIVERSITY OF CALIFORNIA. + +Class + +A blank, light yellow page. + +10- + +A FEW CHAPTERS +IN +ASTRONOMY. + +300000000000000 + +A FEW CHAPTERS + +IN + +ASTRONOMY. + +BY +CLAUDIUS KENNEDY, M.A. + +LIBRARY UNIVERSITY OF CALIFORNIA + +LONDON: +TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. +1864. + +GENERAL + +A large snake with a flame on its back, holding a staff. +ALEXANDER FLANNAGAN. + +PRINTED BY TAYLOR AND FRANCIS, +RED LION COURT, FLEET STREET. + +PREFACE. + +The mathematical discussions in this little book are quite elementary, and geometrical in character, except that, in three instances, two differentiations and an integration of the most rudimentary kind have been used. In a very few cases, the results of analysis have been simply accepted; and even of these, few as they are, some are given only to verify conclusions already arrived at independently. + +November 15, 1894. + +192462 + +ANALYSIS + +The first part of this list is of the nature of a preliminary analysis of the +general character of the various forms of the disease, and includes a general +description of the symptoms, a brief account of the history of the case, and +the results of the treatment. The second part is a detailed analysis of the +cases which have been treated by the author, and includes a description of +each case, with its history, symptoms, treatment, and results. + +I am indebted to Dr. Henry W. H. Smith for his valuable assistance in +collecting and arranging these cases, and I am also indebted to Dr. J. H. +Baker for his kind permission to use some of his cases. + +This list is not intended to be exhaustive, but merely to give a general +idea of the nature and extent of the disease. It is hoped that it may be +of service in directing attention to this important subject, and in aiding +those who are engaged in its investigation. + +1 + +CONTENTS. + +Page + +Preface .................................................. V> + +CHAPTER I. +ON A VIRTUAL ILLUSION AFFECTING CERTAIN ASTRONOMICAL PHENOMENA. + +Difference between spheres of vision and plane of vision—Middle of moon's illuminated limb and the point existing above sun—Deceptive appearance of curvature in meteor path—Danger of making the radiant of a very sparse meteor system higher than the reality—Danger, when using alignments only, of making the position of a very faint object lower than the reality—Possible modification by this illusion, of the apparent curvature of the very long tail of a comet. + +CHAPTER II. +THE EFFECT OF THE EARTH'S ROTATION ON CERTAIN MOVING BODIES. + +Some brief historical notes—Resolution of earth's rotation into $V_r$ or that about the vertical line, as axis, and $M$, or that about the horizontal meridian line, as axis, and their rotationally moving bodies. Some remarks including logarithmic spiral described by homing pigeon over the sea. Funenius' Pendulum postponed to Chapter IV—Effect of $M$ on vertically moving bodies. Body of high density dropped from a height; resistance + +viii + +CONTENTS. + +Page + +of air being taken as unimportant. Experiments of Gugleldini, Benzenerg, and Reich. Body of very low density falling in air—Effect of $M$ on the rate of a perfectly free pendulum—Effect of $V$ and $M$ together. Some interesting new effect on bodies of high density—Effect on projectiles postponed to next Chapter—Notes . . . . . . . . . . . . . . . . 11 + +CHAPTER III. + +DEVIATION OF PROJECTILES FROM THE ROTATION OF THE EARTH. + +This effect is relatively very small—Whole shift of point of fall of projectile from rotation of earth is compounded of three shifts, viz. (a) the (purely) longitudinal shift, (b) the (purely) transverse shift, and (c) the westward shift—Whole aberration of trajectory is equal to sum of these three shifts—(a) and (b) whole deflection is (d) combined with transverse component of (c)—Formulas for the various shifts in terms of $r$, the range, $h$, the height of trajectory, $q$, the time of flight, and $\delta$, the angle of deviation—The formulae for the longitudinal shift are applicable to ballistic to parabolic trajectories; while those for parabolic trajectories in terms of initial velocity, elevation of discharge, and $y$, are altogether impracticable to ballistic trajectories—Tables of deviations—Notes . . . . . . . . . . . . . . . 34 + +CHAPTER IV. + +FOUCALIS' PENDULUM. + +Discussed separately, though belonging to Chapter II—In behav- +iour a dynamical problem, and by no means a mere kinematical consideration—The pendulum is suspended in a mode of suspension, from its own inherent nature, and from resistance of air—Pre-eminently important to keep its amplitude of oscillation, both angular and linear, as small as practicable—Incidentally useful in determining the value of $M$—In his later experiments, Mr. Buns's later experiments specially successful—Notes . . . . . . . . . 60 + +CONTENTS. +ix + +CHAPTER V. +ON THE POSITION OF THE DYNAMICAL HIGH TIDE RELATIVELY TO THE CELESTIAL TIDE—PRODUCING BODY. + +Magnitude of lunar tidal forces.—The tide is a wave ; though a Page +forced one.—Motion of water in a wave ; especially in a tidal wave — Tangential, or horizontal, tidal forces greatly more important than radial, or vertical, ones. The latter complicate with the former the motion of the water, and give rise to many +interesting modifications. The general result is almost as if the forces were wholly tangential ; they shall be taken so—the equation, see $\mathbf{v}=\mathbf{w}$, taken as granted—if undisturbed water be at rest, and if the force be applied perpendicularly to the surface. Still, +the free tidal wave could not keep up with the moon. If it be "deep," that is of greater than the critical depth, the free tidal wave would go faster than the moon—Position, relatively to +which the moon's motion is measured—is then called " shallow." If this +according as water is " shallow," or " deep," and without, or with, +friction. Case of water of critical depth discussed further on— + +The two answers for frictionless water can be given by means of +a system of differential equations. In the first place, when the tide +must be in such a position that the tidal forces shall be working with gravity, so as to accelerate the oscillation of the water, which means that low water is under the moon; and vice versa, for +deep water must be in such a position that the tidal forces shall work backwards, and with " deep" water, backwards—Discussion of case in which the water is of critical depth—Shift, whether forwards or backwards, of high tide by friction is greater as the coefficient of friction is greater; but still, even when friction is zero, it could +shift slightly either way—For this and another independent reason the crest of the dynamical high tide cannot be, under any circumstances, 45° behind the moon—Solar tides—Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 + +CHAPTER VI. +THE " HORIZONTAL" PENDULUM. + +This is a convenient name; though the Pendulum's rod need not, +and its plane of oscillation must not, be horizontal.—Different + +b + +X +CONTENTS. +Page +modes of suspension described, with some reference to their com- +parative advantages.—Mode of obtaining the sensibility of the +instrument without having to depend on the accuracy of working +of the setting screw.—NOTES 93 + +CHAPTER VII. +THE MOON'S VARIATION. +Magnitude of solar disturbing forces producing the Variation—Diagram of the Var.—The Var. in elongation and in radius- +vector—The P.L. orbit, relatively to earth and line joining earth and sun, is a compound epicyclic curve, with deferent and first and second epicycles—Its radius of curvature at syzygies and at quadratures—it is very slightly flatter, at syzygies and at quadratures than at perigee and apogee—Some exceptions apparently new geometrical proof that the tangential disturbing forces, by themselves, would produce an oval Var. orbit with its least axis in syzygies, and that the radial force, by themselves, would produce an oval Var. orbit with its greatest axis by their direct immediate action, only 4/11th of the Var. in elongation ; while the tangential component of the earth's attraction on the moon produces the remaining much greater part—Some mistakes easily made concerning the Var.—NOTES 104 + +CHAPTER VIII. +THE MOON'S PARALLACTIC INEQUALITY. +Magnitude of the solar disturbing forces producing this in- +equality—Diagram of the P.L.—P.L. in elongation and in radius- +vector—the P.L. orbit, relatively to earth and line joining earth +and sun, is a compound epicyclic curve, with deferent and first epicycles at conjunction and at opposition—The existence of this inequality pointed out, and its magnitude estimated, by Newton ; though it had not yet been deduced to his knowledge by observation—Some apparent anomalies in the P.L. orbit show that the acceleration and retardation of the moon's motion are always + +A diagram showing the Moon's parallactic inequality. + +CONTENTS. +xii + +Page + +contrary to what the solar tangential forces are endeavouring to produce—While the system of disturbing forces causing the Var. has two axes of symmetry, one in syzygies and the other in quadratures, that causing the P.I. has only one axis of symmetry, a geometrical dynamical difference—in the P.I. orbit the tangential component of the earth's attraction on the moon is never less than 3:31 times as great as the opposing solar tangential component at points of elongation greater than this—Thus the immediate cause of the P.I. in elongation is the earth's own attraction.—Notes +132 + +A blank page with a light beige background. + +UNIVERSITY OF CALIFORNIA + +A FEW CHAPTERS + +IN + +ASTRONOMY. + +ERRATA. + +Page 25, line 6 from bottom, for FHD read FHD. +" 30, line 8 for the Hf read DF, +" 30, line 3, for a reast C, twice, +" 32, line 7 from bottom, for reast r, +" 36, line 15 (Nore) I read C, +" 130, Nore G... The relative reduction of the sun round the earth would prevent the ellipticity from increasing beyond a certain limit. + +But ordinary observers, and even astronomers themselves, are not in the habit of referring to such objects as those around them to the sphere of vision. Such objects are referred to what writers on perspective call the plane of vision at right angles to the line of sight, which the eye, as it were, always carries about with it. There are different reasons for this. The idea of the plane of vision is, in some respects, simpler than that of the sphere of vision, and perhaps more nearly applicable to the observations made most readily as it is always at a constant angular extent. Besides this, all ordinary drawings and pictu- +tures are made on plane surfaces, for different obvious reasons. + +B + + + + + + + + +
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+ +A vintage document with text in a formal script. + +A FEW CHAPTERS +OF CALIFORNIA + +A FEW CHAPTERS + +IS + +ASTRONOMY. + +CHAPTER I + +ON A VISUAL ILLUSION AFFECTING CERTAIN +ASTRONOMICAL PHENOMENA. + +In considering the visual positional relations of the heavenly bodies to each other and to various given lines and planes, it is of course necessary to regard them as projected on an imaginary spherical surface whose centre is at our point of vision. + +But ordinary observers, and even astronomers themselves, are not in the habit of referring terrestrial objects around them to the sphere of vision. Such objects are referred to what writers on perspective call the *pleas* of vision at right angles to the line of sight, which in our eyes is usually always carried out with great accuracy, different results being the consequence of this. The plane of vision is, in some respects, simpler than that of the sphere of vision, and presents itself more immediately to the observer; and this the more readily as it is always of limited angular extent. Besides this, all ordinary drawings and pictures are made on plane surfaces, for different obvious reasons. + +2 + +2 +ON A VISCAL ILLUSION AFFECTING + +When the eye is stationary the angular extent of distinct vision is quite small. Even if the eye be allowed to move, while the head still remains stationary, the angular range of vision, or the extent of the field of view which can be attained without too much disturbing effort, though much greater than before, is still insufficient to include in each case the difference between the plane of vision and the sphere of vision may be practically of very little importance. + +But it would be otherwise if the plane of distinct vision could be made larger, for than its own perspective would amicably affect the question. We need not, however, go into this; for if we would compare two objects whose horizontal angular distance is too great for them to appear in the same limited field of view at the same head position, such as a body round a vertical axis from one to the other; and we shall by turns in its own separate limited plane of vision, and usually with a very indistinct idea of the geometrical relations between those different planes. This is the main cause of the illusion now in question. + +We have just mentioned here the most striking instance of such illusion, referring to it further on, for explanation. It has the advantage of being easily observed every month. If the crescent moon be not less than two or three days old, the sun being near setting, the middle of her illuminated limb, which, of course, is turned directly towards the sun, will seem to point decidedly above the sun. As the angular distance of the moon from the sun increases, the apparent discrepancy becomes more marked, until at last it becomes so great as to be beyond comprehension. Physicists and mathematicians, who are course are perfectly aware of the real conditions of the case, will acknowledge that they cannot direct themselves of the feeling that they actually see the middle of the moon's illuminated limb to be pointing several degrees above the sun. + +The unequal raising of the sun and the moon by refraction has no share in producing the illusion; for its effect is to raise + +CERTAIN ASTRONOMICAL PHENOMENA. + +the sun, which is at the horizon, more than it does the moon, which is higher in the sky, and so to diminish the illusion to a slight extent. + +Every great circle of the sphere of vision is, to the eye viewing it simply and unconsciously, a right line, because the eye is in the plane of it. The horizon is such a great circle, and it is to the eye a right line ; a straight edge held in the hand can be applied to it and will fit at any place. Not only this, but the horizon presents itself to the mind as everywhere a right line ; for a reason which we shall mention presently. + +But if we could draw any other great circle on the sphere of vision, or even one passing through its center, it would be to the eye viewed it unconsciously perfectly straight in every part, it would intersect the horizon in two opposite points, while having some elevation above the horizon at its middle part; therefore to the mind of the beholder, who is so habituated to dealing with lines &c. as drawn on plane surfaces, and knows that two right lines, as existing on such surfaces, cannot meet to form a point, he would conclude that his perception of its inclination to the horizon would be continually varying, as it was followed by the eye from end to end. A person standing opposite the middle of a very long, straight, horizontal, architectural feature, or other such line, of sufficient height, can only with difficulty divert himself of the idea that he as turns his head from side to side, he sees that line as a curve with its convexity towards him when looking down upon it, and towards him when looking up upon it; but on each side as sloping down to the horizon. + +If we suppose a straight line to be traced on the sky at a considerable altitude, we should not refer its direction, at any point, immediately to the horizon, perhaps quite out of sight. But we should do what would be equivalent; that is, refer its direction, at any point, to the vertical at that point; and we always do perceive this fact in all our observations from the sense that we have of the direction of gravitation. As we + +n 2 + +4 +ON A VIRTUAL ILLUSION AFFECTING + +consider, in succession, at not too great an altitude, the imaginary vertical great circles which we trace for ourselves in succession while turning the body round, we take them as parallel to each other, because they have been similarly related to the successive outlooks. But they are actually converging; the consequence is that a straight line or great circle traced across them would cut out at very peripetiously different angles; even within a circumference, two points on such a different angles lines which in one sense have the same direction, or face each other in their own fashion, being all at right angles to the horizon, and therefore it seems to be curved. + +But why does the horizontal great circle look straight, whilst another inclined to it looks curved; both being straight? The reason is that, in looking round, the observer turns his head, or rather his eyes, so that he always faces the horizon. Hence the horizon has the same directional relation to his outlook, as he faces it standing erect. The horizon, therefore, naturally becomes the most general line, or plane, to which the position of an object is referred. But this is not the ease with the other great circle. As the beholder turns round on his vertical axis, which is inclined to the plane of that great circle, every point of that plane will appear to be referring itself to his outlook, as he stands erect; consequently when the eye is made to run round it in the ordinary way it seems curved. + +But if, while holding himself up straight, he were turned to a post which then was inclined until it was at right angles to the plane of the great circle now in question, and if the post were then rotated on its axis, the familiar horizon being hid from view, then he would see that he could not possibly turn him to be straight, for the same reason that the horizon ordinarily does so; and if he fixed his attention very strongly on that line, then the horizon, if uncovered, would seem to him to be curved as he was being turned round. That this would be so can be proved by an easy experiment. Stand near the corner of a long enough room, or lobby, or passage, and view the opposite cornice. + +CERTAIN ASTRONOMICAL PHENOMENA. +5 + +or the juncture of wall and ceiling, running along the length of the room &c. The cornice will be everywhere straight to the eye; yet its visual inclination to the horizon increases con- +tinually as the eye follows it from the nearer to the further end. Hold a straight rod vertically, with both hands, so as to visually cross the observer; now while the arms remain rigid and move only under the direction of the eyes, slowly turn +the whole body from side to side briskly, keeping, with the eye and attention strongly fixed on the visual intersection of rod and cornice; and the cornice will appear to be curved with the concavity downwards, on account of the continual change of its visual angle of intersection with the vertical rod. A few trials may be necessary in order to catch the effect, as it is considerably difficult to keep the eye steady when the line is straight, and by the observer's involuntarily comparing consecutive portions of it with each other; which latter cannot be done in the case of an imaginary straight line on the sky. + +To return to the case of the crescent moon referred to already. Suppose that our latitude is nearly that of London, say 51°1', and that the young moon is 45° from the sun, which is setting; and, to obtain a mean condition, let the time be an equinox and the moon on the celestial equator; the altitude of the moon will be 20°. If the straight line along which the middle of her illuminated limb points towards the sun could be traced on the sky, its inclination to the horizon would be, of course, at the sun 28°1' (the completion of the latitude); but at this moon it would be 37°1'. This difference is greater than any observed in nature than at the moon by 9°. If persons who had not considered the matter, and even some others, when off their guard, tried to trace that line by the eye, they would start from the moon at a downward slope of 29°1', and preserve that slope as well as they could, until reaching the horizon; just as they would do if dealing with a straight line on a plane surface directly facing them. This of course will carry them many degrees above the + +6 + +ON A VISUAL ILLUSION AFFECTING + +sun. But if the observer were in some unaccustomed attitude, +say half reclining and looking obliquely over his shoulder, so as +to obscure his sense of the vertical or horizontal direction, and +if all known horizontal and vertical lines were properly con- +cealed from view, and if he had a good eye for straightness and +symmetry, he would doubtless be able, having started in the +proper direction from the moon, to continue his trackless course +until hitting off the horizon. + +Perhaps the simplest, and for some persons the most striking, +exhibition of this deception would be when the moon is in the +first quarter, or "half moon," and the sun is setting. Suppose +the altitude of the moon to be, at the time, $m$ degrees. The +terminator, or boundary of light and shade on the moon, is +straight and vertical, and the middle of the illuminated limb is +projecting vertically upwards from the horizon. At this time the +setting sun degrees lower over. If we try to follow by the eye +the direction in which it points, we shall be tempted to trace for +ourselves an imaginary line on the sky everywhere horizontal +and having always the same distance from the horizon, as we +should do in a diagram on a plane surface ; and the result will +be that our production of a line, which really points directly at +the sun's position, will be a circle (Sun's line). If traced on the sky, +it would be a small circle of the celestial sphere, and, paradoxal as it sounds, everywhere convex towards the straight horizon.) + +In this case the illusion is obvious, and felt at once to be +something that requires explanation ; besides which it is not +calculated to lead to any ulterior mistake. + +But there is another exhibition of this illusion which is not of +so innocent a character; it does not manifestly betray itself as +an illusion, and it has given rise to misconception. It is a +seeming phenomenon which by ordinary persons is not con- +sidered to require explanation, because it appears at first sight to +depend so evidently on another principle. Even those who + +CETAIN ASTRONOMICAL PHENOMENA. +7 + +must be aware of the actual circumstances in this case, have not, +so far as we know, given any warning on the subject, at least in print. + +Every one must have noticed what seems like the well-marked curvature of the path of an ordinary meteor or shooting-star, whether a sporadic one, such as may be seen on every clear night, or one belonging to a shower, provided its apparent path did not happen to be exactly parallel to that of the sun. The way in which the present writer can testify, very strikingly displayed (if this be not a bully) by the shower of Andromedids *, or Bicilide, on November 27, 1872. Certain others also remarked the same, as anyone must have done. Any pictures (not diagrams on a star map) that we see of meteor showers invariably give a decided curvature, curve downwards, to the luminous tracks. To us it seemed impossible that any meteor could do so; the contrary. One of the observers just referred to, speaking of that shower of Andromedids a couple of days after its occurrence, remarked how interesting it was to see the curvature of the trajectories of the celestial projectiles due to gravitation. + +But a moment's consideration will show that this is quite a mistake. The most point of any of these visual tracks was visibility no less than their length. The distance between the object itself being many miles in length. Now the very longest period of visibility that we can allow to any of those meteors is two seconds, in which time one of those bodies would fall, considering the resistance of the air, less than 64 feet. But a linear deflection of 64 feet would be quite inaccessible to the eye in such luminous tracks as these. It is only when they are seen at right angles to the apparent track, which will but seldom happen. The case, of course, is quite different of a large meteorite which is seen by an observer to fall to the ground, net far off, after having been visible for a longer time. The illusion now in question is clearly due to the constant change of the + +* These are sometimes called "Andromedids," as though the name of the constellation were Andromo. + +8 + +ON A VIRTUAL ILLUSION AFFECTING + +inclination to the horizon of the seemingly straight luminous tracks of the meteors. + +It is true that, unless the direction of the motion of a meteor is parallel to that of the earth, when the meteor enters the earth's atmosphere the resistance of the air will not only pro- +duce a violent retardation of its velocity, but will cause a deflection and change in its course, which will necessarily be fixed and definite. But this curvature will not be visible to the observer. +This is easily seen thus: Suppose the meteor to be visible, even before entering the atmosphere, the observer would see only its motion relative to the earth, the air, and himself, all regarded as stationary ; when the meteor, with this apparent motion, enters the apparently stationary atmosphere there is nothing to cause any perceptible change in the position of the observer ; no curva- +ture whatever of the sphere will be produced by the resistance of +the air in the path of the meteor. (Nor will there be any +change in the position of the apparent radiant produced by said +resistance. We mention this because the contrary has been +directly contended for.) + +The reason why the seeming curvature in a meteor's track is +not greater than it seems to be this, that the eye is not only +compacting the track with the vertical, or the horizontal, at every point, but it is also to some extent comparing contiguous lengths of +the track with each other; and this tends to correct and diminish the illusion. + +For this reason the more rapid the flight of the meteor, the +less will be the appearance of curvature in its path, for in such +cases when it approaches more nearly to the condition of a +luminous line seen at once from end to end, the parts of which +can be more readily compared with each other. This was well +illustrated by many of the quick-moving Persids of August 10, +1883.* + +* There is a detail of this illusion which is worth mentioning. It appeared to be very noticeable with a large proportion of the meteors of Nov. 27, 1872. Near the end of visibility, the apparent downward curva- + +A diagram showing a meteor entering Earth's atmosphere. + +CERTAIN ASTRONOMICAL PHENOMENA. +9 + +The illusion of which we now speak may easily lead some persons into error when endeavouring to fix upon the radiant point of a very sparse meteoric system. + +If the insufficiently-experienced observer has not been for- +tunate enough to catch with his eye any of the few meteors nearly near to the radiant point, he will, in producing the visible parts of the meteor-track backwards, almost certainly pass above the point at which so fix the position because it is too low. Or, if on the left or right of the object belonging to a certain known radiant, he might easily refer thereto some sporadic meteors really coming from a different origin at a lower altitude, when perhaps it might be important to know that, in fact, none belonging to the radiant were to be seen on that night. + +Conversely, when endeavouring to fix the position of a visible very near radiant, he may easily pass below its points with known stars at considerable angular distances from the object, he may easily do the opposite; that is, assign to it a position lower than the true one. From his alignments the very faint object might be found again on the following night by himself, though perhaps not by another, whose skill in allowing for the illusion now in question might be either greater or less than his. + +This illusion might, with some persons, slightly affect the apparent curvature of a comet's tail, if very long. Some years' course of the comet's path seems to increase somewhat rapidly, as in the ballistic trajectory of a projectile, caused by the resistance of the air. This also is represented in some pictures of meteoric showers. But though gravitation tends to reduce this curvature (which is only slight), yet still it is utterly impossible for the reason given above that it could have been perceptible to the eye. The deception may be due, in some way, to the fact that the eye is following the apparently curved path of a luminous particle through a medium which is not perfectly transparent. This effect is more rapidly retarded. This seeming phenomenon gives rise to another mis- +apprehension. It makes the meteors look at the end of their luminous tracks, as though they were no farther off than the falling stars of a rocket. + +10 +**VISUAL ILLUSION AFFECTING ASTRONOMICAL PHENOMENA.** + +ago there was a difference of opinion between two correspondents in a popular scientific periodical respecting the curvature of the (long) tail of the great comet of 1852. This was, in all probability, produced by the cause above mentioned. This might be not unimportant, in view of the conclusions as to the composition of comets, which have been drawn from observations of their curvature in connection with the known motions of the comets. But a comet's tail, being a visible and permanent object during the observation, so that different parts of it can be compared directly with each other, is much less liable to be affected by the illusion now in question. + +[ 11 ] + +CHAPTER II + +THE EFFECT OF THE EARTH'S ROTATION ON CERTAIN MOVING BODIES. + +It was believed by Aristotle and by Ptolemy that the earth's rotation, if it existed, should affect the motion of certain freely moving bodies. Galileo also perceived that this must be so, while rejecting the particular effects contemplated by them, at least as regards the falling body. Newton was the first to point out that freely falling bodies must deviate to the east of the vertical, on account of the rotation of the earth ; and he suggested that experiments should be made with these in order to obtain direct proof of that rotation. Such experiments were tried by Hooke, in 1680, but with an insufficient height of fall. In 1836 Edward Stoney, C.B., of Edinburgh, showed that the earth's rotation could be demonstrated by means of what is now well known as the Foucault pendulum. This experiment thitherwith. In 1837 the subject was discussed, in connection with the flight of projectiles, by Poisson. It came much more prominently before the general public when Foucault exhibited his famous Pendulum to the French Academy in Feb. 1851. Shortly afterwards he devised, for himself, and actually performed, an experiment with the gyroscope which had been proposed fifteen years before by Sanson. + +A common and popular explanation of the deflection of projectiles, currents of air, &c., from the rotation of the earth, is that if, in our N. latitudes, a body be moving southwards it is all the while passing ever ground which has a greater velocity eastwards, from the rotation of the earth, than the + +12 +THE EFFECT OF THE EARTH'S ROTATION + +ground which it has started from, or has lately crossed, and that therefore it is left behind a little towards the west, or the right hand, by the surface of the ground beneath it; and that, for corresponding contrary reasons, when moving northwards it will gain on that surface towards the east, or still to the right hand. This is, of course, perfectly true; but the particularities of meridional motion are not so interesting as they appear, what is sometimes directly declared, viz., that the above statements are not applicable to bodies moving in the east and west direction. It is strangely forgotten that if a point on the solid ground south or north of an observer is moving towards his left, when he faces it, relatively to him as centre, with a certain angular velocity, a point on the ground east or west of him must be doing the very same, and that this point may be free from deflection whatever its latitude may be left behind, to the right in $N$, and to the left in $S$. Latitudes, in whatever azimuth direction it may be going; and that, other things being equal, its apparent deflection must be the same for all azimuths of motion. + +The period of the earth's rotation is, of course, a sidereal (not a solar) day; this contains 86161 seconds of mean solar time. The mean declination of the sun at noon varies from $0^\circ$ to $360^\circ/86161$, or 15.04 seconds of arc, which in circular measure is $2\pi/86161$, or $1/3.1731^{\circ}$; thus then represents the earth's angular velocity of rotation, which we shall denote by $\omega$. + +The resolution and composition of rotations is among the first elements of rigid dynamics. The two components of the earth's rotation with which we are now concerned are $F$, or that about the vertical line at the locality in question as axis, whose angular rate is $\omega\sin\lambda$, $A$, being the latitude of the place, and $M$, or that about the horizontal meridional line at the locality as axis, whose angular rate is $\omega\cos\lambda$. (See Note A.) + +*It is interesting to note that 15.1731° itself, the magnitude of its own logarithm to five decimal places. But we need not attach any mystical significance to this coincidence.* + +ON CERTAIN MOVING BODIES. +13 + +We may give here a practical illustration of the existence of these two components of the earth's rotation. If in N. latitudes a star close to the horizon be observed with a telescope whose eye-piece is furnished with a micrometer scale, the star will be found to have a motion in the horizontal direction towards the right (whatever vertical motion it may have compounded therewith); and this horizontal motion will be found to be the same for all stars at any given latitude, in whatever direction they may be; and the angular rate of the horizontal motion will prove to be that of the earth's rotation multiplied by the sine of the latitude; this being due to the earth's component rotation $F$. Similarly, if one observes any stars close to the prime vertical, or the great circle passing through the zenith and the E. and W. points on the horizon, he will find that they all have the same rate of motion towards the left (whatever vertical motion they may have compounded therewith); and this angular rate of motion along the prime vertical will prove to be that of the earth's rotation multiplied by the cosine of the latitude; this being due to the earth's component rotation $M$. + +Of sufficiently free bodies, those which are moving horizontally are affected by the component $F$, by which the surface of the ground at the place of observation rotates in its own (instantaneous) phase. Those which are moving vertically, whether upwards or downwards, are affected by the component rotation $M$, by which the surface of the ground is always being tilted over eastwards. + +We shall first consider those which are influenced by the component $F$. It may be best to begin with an imaginary ease, for illustration. Suppose a body started to slide on a perfectly frictionless, even, horizontal surface, or floor, in a vacuum. If the floor were stationary the body would, of course, describe, from here to there, a straight line through with uniform velocity. But as the floor is always rotating in its own (instantaneous) phase + +14 +THE EFFECT OF THE EARTH'S ROTATION + +with the angular velocity $\omega$ sin $\alpha$, and as there is no connection between the floor and the body which would make the latter partake of the rotation, it will not do so; but it and its radius-vector will be left behind, and that time, if visible, would appear to rotate about the point of starting, watch-wise in N. latitudes, and anti-clockwise in S. latitudes, with the uniform angular velocity $\omega$ sin $\alpha$, while being itself described by the body with the uniform linear velocity $v$. Consequently the body would describe about the point of starting, as pole, a spiral of Archimedes, whose equation would be $r = \frac{v}{\omega} \sin \theta$. If the body were started from the middle of the floor with such a small velocity that it would not reach the edge of the floor for a few days, it would present the curious phenomenon of revolving (with an ever widening orbit) round the point of starting for no apparent reason. + +We must, however, content ourselves with the consideration of masses moving horizontally under more ordinary conditions. + +The winds afford a familiar instance. The explanation of the direction of the trade-winds and cyclones is now pretty generally understood. It is due to the fact that on account of the heating of the air by the sun in the neighbourhood of the thermal equator causes that air to ascend, which occasions an indraft of the lower air both from the N. and from the S. For a non-rotating earth, the general direction of these would be meridional; but the rotation of the earth causes an apparent turning to the right on the north side, and to the left on the south side, of the greater part of them; so that those coming from the north side, and the S.E., trades on the south side, of that line. + +A sufficient local extra heating of the air in N. latitudes causes, in a similar way, an indraft of the lower air from all sides; the component rotation $T$ causes the converging masses of air to pass in N. latitudes to the right of the centre of the super-heated area, which produces a vortex turning in the opposite direction; and this vortex is carried along with it up upon the table; the corresponding result in south latitudes being a vortex + +ON CERTAIN MOVING BODIES. +15 + +turning the other way, or with the hands of the watch. Such vortices being called cyclones. + +Ocean currents must be very considerably affected by compo- +nent rotation $V$; but these are subject to a variety of other +important influences of which we shall mention only prevailing +influence, viz., that of the earth's rotation. It will generally be +generally impossible to distinguish the effect now in question from +others, and useless to speculate therefore; except perhaps +in one apparently simple instance, with which, as it happens, +we are practically concerned. It can hardly be doubted that it is +largely in consequence of component rotation $V$ that the warm +Gulf Stream bears so strongly on the coast of north-western +Europe, and that it is worth while to advert to this following --- +There are five great ocean currents. The two in the Indian +viz., that in the N. Pacific and that in the N. Atlantic, both turn +watch-wise. The three in S.E. latitudes, viz., that in the S.E. +Pacific, that in the S. Atlantic, and that between S. Africa and +Australia, all turn counter-watch-wise. It seems highly +probable that all this is, at least, promoted by the earth's +rotation. For example, the Gulf Stream is due to a preferential +current movement of the water, produced somewhat under the +condition of the earth's rotation, the direction of its turning +would be such as we have just mentioned, and opposite to that of +a cyclone in the same latitude produced as above described. + +The course of the flight of migrating birds is probably some- +times affected by component rotation $V$. But as the consideration of A. H. Tait shows, it is not necessary to confine ourselves another one which might actually occur.--The keeper of a light-house several miles out from the coast has some homing pigeons, bred by himself, which are well acquainted with the district. One is let go from a point on the coast; it starts at once to return directly to the light-house; the bird is guided solely by his sight of the light-house, and the water being per- +fectly still, he does not even attempt to follow the line of course edging sideways to the right of the instantaneous straight line + +16 +THE EFFECT OF THE EARTH'S ROTATION + +from himself to the light-house; he will keep his head always pointed directly towards the light-house, and to do this he must be continually turning very slowly towards the left, doubtless without perceiving that he is doing so. The forward velocity of his flight is uniform, and his involuntary sideward motion to the right will go on increasing, until at the resistance of the per- +fectly air tight case, it becomes great enough to prevent any further increase therein; then the forward velocity which has reached its final magnitude, and becomes constant, like the forward velocity. The bird's visible course, or that relative to the surface of the earth, will then become $\text{quan} \propto v$, a loga- +rithmic spiral described backwards towards its pole, which is at the light-house (See Note B). + +A. At the pole, when this has taken place. If now the latitude be $50^{\circ}$, that of London, and the distance from A to the light-house be 10 miles, and the bird's velocity of flight be at the mean for such cases, say 800 yards per minute, or 40 feet per second, and his weight 14 oz., and the coefficient of adeward shifting $0\cdot5$, which we have good reason to believe is pretty nearly correct; then his greatest departure to the right of his initial course will be about 3 miles, and his greatest deviation just about 70 yards, at the distance of 3-68 miles from the light-house. If this departure seems somewhat small, let us remember that it has taken place in spite of the bird's constant (unconscious) endeavour to avoid it, and in spite of the lateral resistance of the air. + +Probably there is always a sensible deviation of this kind when a bird is travelling to a sufficiently distant intended goal. His difficulty in avoiding it would depend upon whether one of the more prominent objects in view would make him more or less aware of his sideward shifting, and thus suggest to him to make some allowance for it by directing his head to the proper side of the goal, or the left in N. latitudes; but the amount of angular allowance necessary would depend on the velocity of flight, and on the latitude, and also on the bird's own weight and his coefficient of sideward shifting; and it seems very unlikely that + +ON CERTAIN MOVING BODIES. + +Instinct, much as it can do, would enable him to make due allowance on account of these; though it would doubtless enable him to provide against a cross wind. + +We see, then, that the familiar expression, "As straight as the crow flies," should not be lightly used, without distinctly postulating the condition that the bird is making due correction for the rotation of the earth. + +In the case of a railway engine and train running along a perfectly level line, the rails being perfectly level with each other, the sideward shift is prevented by the resistance of the right-hand rail in N. latitudes, and of the left-hand rail in S. latitudes; and it is said that the right-hand rail and the flanges of the right-hand wheels get more wear in N. latitudes, on this account, than the others. + +This is undoubtedly so; we know already (see Nort) that the expression for the pressure $P$ against the right-hand rail is + +$$P = \frac{2\pi v^2}{g}W \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)$$ + +in which $v$ is the velocity in feet per second and $W$ the weight of the moving body (see Nort). For an engine going at 30 miles an hour, or 44 feet per second, in the latitude of London $51^\circ 20'$, this would be 1/610th part of its weight, and if this weight were 90 tons, the pressures would be 10-48 lbs., and they would be distributed equally all over the right-hand wheels. The effect, then, is so small that it must be undistinguishable; as it would be altogether overborne and masked by a gentle cross wind, or by a difference of level between the rails, say ±71 feet apart, of only 1/100th centimetre; or by a gentle curve in the line of 000 miles radius; not to mention other causes of unequal wear. + +It is said also that rivers in N. latitude must, for the same reason, wear away their right-hand banks slightly more than their left ones. This is undoubtedly true; but the effect is utterly imperceptible; not only because the greatest velocity of a river flow is relatively so small, but also on account of the + +c + +THE EFFECT OF THE EARTH'S ROTATION + +insensibly greater effect of various other causes of inequality in the cession of the banks. + +Among moving bodies influenced by $Y$ must be mentioned the famous Foucault's Pendulum; but this is deserving of a chapter to itself, which we shall give it. + +We now come to moving bodies which are affected only by the earth's other component rotation $M$, that is to say those moving vertically, whether upwards or downwards. Just as in $N$, latitudes, a sufficiently free body, projected or moving away horizontally from before a spectator standing vertically, will deviate towards his right, so gravitation would be transmitted to a spectator lying horizontally in the meridian in $N$, latitudes, with his feet to the south, a body projected away from before him in the plane of the prime vertical will deviate to his right, owing to the rotation of that plane in itself with the angular velocity $\omega \cos \lambda$. Gravitation alters the case, except for a very slight difference between the two cases. For when a body discharges the projectile vertically upwards its deflection towards his right is one to the west. If he lie face downwards, say at the edge of a mull cliff, and discharge, or simply drop, the body downwards, its deflection towards his right is one to the east. + +We shall here discuss the latter via, a body dropped from a height. That such must deviate to the east of a thumb-line is easily seen otherwise thus. A point on the surface of the ground is moving eastwards, from the rotation of the earth, with the linear velocity $R\alpha \cos \lambda$, $R$ being the earth's radius ; but a point directly over it, at the height $h$, is moving eastwards with the velocity $(R+h)\alpha \cos \lambda$. The latter is therefore moving eastwards faster than the former with the additional velocity $h\alpha \cos \lambda$; consequently it moves simply towards the east. But if a body falls with its fall, have left the lower point behind it towards the west. This deviation, $\delta$, of the body towards the east, if the resistance + +# UNIVERSITY OF CALIFORNIA + +## ON CERTAIN MOVING BODIES. + +19 + +of the air to the falling body be neglected, is given by the equation + +$$\ddot{z} = \frac{3}{2} \frac{d^2 z}{dt^2} \cos \lambda, \quad \ldots \ldots \quad (2)$$ + +which is, for same $\lambda$, one fourth of the *Westervet Shift* in the Chapter on the Deviation of Projectiles (since, for same $h$, this $t$ is half the other). [For proof see Note C.] It being in a vacuum $g = 4 g_0$, and the above expression for this deviation can be written $\dot{y} = \sqrt{\frac{2 h}{g}} \sin \lambda$, as it usually is. + +A body dropped from a height must have also, as is evident, and as was pointed out by Hooker, a very small deviation towards the south; $\dot{y}$ is not produced by $M$, but by the horizontal (southward) component of the centrifugal force of the earth's rotation being greater at the top of the height of fall than at its bottom (the vertical component of which is zero). Its magnitude, which is easily obtained geometrically, is only $\frac{h}{g} \sin 2\alpha$, or $\frac{h}{g} \sqrt{\sin^2 2\alpha}$; neglecting the resistance of the air. The presence of $\dot{y}$ in it shows, at a glance, that it must be excessively small for all practicable experiments. In that of Guglielmini, mentioned below, it would be less than 1/50,000th of an inch. [The experiment was made by Prof. Bartholomew Price obtained analytically.)] If, in the analytical discussion of the deviation of a falling body from the vertical, quantities of higher (i.e., smaller) orders of magnitude than the first are neglected, this component of it does not emerge into view. A body projected vertically upwards is not affected in this manner, either in its ascent or descent. + +Experiments on this point carried out by various persons to detect the deviation of falling bodies from the vertical owing to the rotation of the earth. For instance by Guglielmini, in 1792, in a tower at Bologna (lat. 44° 30'), with a height of fall of 241 feet; by Benzemer, in 1803, in a tower at Hamburg (lat. 53° 35'), with a fall of 254 feet; and in 1804, in a coalmine at Schlebusch, Westphalia (lat. 51° 25'), with a fall of c2 + +20 + +THE EFFECT OF THE EARTH'S ROTATION + +262 feet; and by Reich, in 1832, in a mine near Freiberg, Saxony (lat. 50° 53'), with a fall of 488 feet. These experi- +ments, especially those of Reich, were, as far as regards the +eastward deviation, satisfactory, considering the delivery of +their nature and the great difficulty of avoiding various causes of in- +accuracy, some of which could produce disturbances often very +much greater than the deviation to be determined. (See Note D,) +it is to be expected that in a full vacuum, the +eastward deviation will be greater if the time of falling can be +made so in a proper manner. Therefore, for given h, the east- +ward deviation is greater in resisting air than in a vacuum. +(But we shall find in Chap. III. that this last is not the case +with the westward deviation of the point of fall of a body +discharged vertically.) +This fact makes the more convenient method of carrying out such experiments as the above. By making the falling body descend +slowly enough we can obtain an eastward deviation, $\delta$, large +enough to be satisfactorily determined, with very much smaller +heights of fall than those mentioned above. The falling body +might be a sort of parachute, very easily designed and con- +structed, which, like a shuttlecock, would be kept rotating about +its axis during its descent. If it were found that it descended +so that it would descend with a uniform velocity, v, of three feet +per second, it would have, with a fall of only 80 feet in +the latitude of London, a deviation, $\delta$, to the east of just over one inch, allowing a little for the lateral resistance of the air. This +deviation is 17 times as much as if the fall had been in a vacuum, +and probably 14 times as much as that of a bullet let fall in air. +In this case the equation + +$$\frac{h}{v} = \mu \omega \cos \lambda; \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\ t = h/v,\text{this value of }t = h/v\omega\cos\lambda.\text{ Therefore, for the}\\ same parachute, the deviation varies as }k^2.\text{ Observe the last paragraph of Note E.}$$ + +ON CERTAIN MOVING BODIES. + +The parachute might be an inverted cone, about three inches in diameter, composed, say, of tracing-paper, and furnished with two very small wings opposite each other and set obliquely so as to cause rotation. If the vertical angle of such cone be 90°, or a little less, it will descend steadily with the velocity mentioned. It should, of course, be made to descend within a chimney-like box, to protect it from movements of the air ; and this should be in a suitable place inside a building ; so that there might be convenient access to it. The parachute would have the qualities of temperature on different sides of it. It would probably be impossible, except by accident, to make the parachute so symmetrical about its axis that it would not be slightly deflected from its proper line of fall by the resistance of the air. But because of its rotation it would descend in a cylindrical helix of air around itself. This would be visible to one who stood at the point of descent and the actual line in a vacuum. If a large enough number of experiments were instituted, in which the parachute was made to start with the same side in different azimuths, the small errors arising from the semidiameter of the helix would be self-compensating. The very small lateral resistance of the air would, of course, slightly diminish the lateral deviation from the rotation of the earth. + +If free fall is one free to swing in any direction, like Foucault's Pendulum, and unlike a knife-edge pendulum, or that of a clock) is affected, as to its rate of oscillation, by its sharing in component rotation $M$, It is, whether it be hanging at rest, or oscillating, rotating about the meridional horizontal line through its point of support, with the angular velocity $\omega_0$-v. There is no centrifugal force acting upon it, as a whirl exists in it, acting on the pendulum at its axis of rotation, which taking the mass as unity, is $\omega_0^2\cos^2\alpha$; r being its mass-radius, or distance of the centre of mass from the axis of rotation. It is evident that if the plane of vibration be $E$ and $W$, this centrifugal force, though apparently conspiring with $g$, will not increase the rate of vibration, because it is always directed + +22 + +THE EFFECT OF THE EARTH'S ROTATION + +along the pendulum rod; it is not parallel with the direction of $g$, except incidentally, at the instant when the pendulum is at the lowest point. Consequently the period of a free pendulum swinging E. and W. is not affected by its rotation with $M$. + +But if the plane of swing be in the meridian, the centrifugal force due to the rotation of that plane about the horizontal $N.$ and $S.$ line through the point of suspension is always parallel to the direction of gravity, and therefore acts vertically downwards; except at the instant when the pendulum is at the lowest point. It is always proportional to the distance of the centre of mass from the said axis of rotation ; but if the amplitude of swing of the pendulum be very small, as it ought always to be in the scientific use of the pendulum, this never differs sensibly from $r$. The pendulum, therefore, is oscillating, not simply under $g$ acting on its centre of mass, but also under a constant, con- spiring, and sensibly constant centrifugal force $\rho\omega^2c\lambda$, acting at the same centre (the mass is still taken as unity). It is very easy to see that if the plane of vibration be not in the meridian, but inclined thereto at the azimuth angle $\varphi$, we shall have for the time $t$ of the vibration of the free pendulum, not + +$$t = \sqrt{\frac{1}{g}} \cdot \text{but (see Note F)} -$$ + +$$t = \sqrt{\frac{1}{g}} \cdot \left( 1 - \frac{r\cos\varphi}{g} \right) \cos\lambda \cos z \cdot q_{\text{mean}} \approx . . .$$ + +(4) + +If then the free pendulum's radius of oscillation $l$ be that of a seconda pendulum, it will gain, in consequence of its own rotation with $M$, $\rho\omega^2c\lambda$ cos $\varphi$ z. in every swing. It being a Foucault's Pendulum, its plane of vibration will rotate relatively to the material surface of the ground once in 24 sidereal hours /sin $\lambda$. Therefore its rate of gaining is constantly varying from zero to its maximum, and back again, with a period of 12 sidereal hours /sin $\lambda$. + +If the pendulum is oscillating meridionally at the equator + +ON CERTAIN MOVING BODIES. + +(Where it will retain its azimuth of oscillation), so that the gain shall be greatest, and if $r$ be 37 inches, which is perhaps a fair mean value of it, the gain will be at the rate of one second in 125 years. Of course the practical unimportance of this does not detract from its dynamical interest. At the poles of the earth, where $c_0$ vanishes, the vibration period of the free pendulum is unaffected by the rotation of the earth. + +We now come to moving bodies which are affected by both components, $F$ and $M$, of the earth's rotation. + +Some of the movements of the atmosphere and of the ocean must be modified by $F$ and by $M$ at once; each making its own special contribution to the whole effect. + +There is a phenomenon which must be largely due to both components of the earth's rotation acting together as auxiliaries. There would appear to be, in equatorial regions, a continuous current from E. to W., in the upper parts of the atmosphere, at the height of 20 miles or so. The peculiar sunsets which began with the great eruption of Krakatoa, in 1883, passed hence successively westward round the equator. It was evident that such an effect could only be produced when a current was travelling in the direction mentioned. Before it became too diffused and widely spread, several passages of it round the equator could be distinguished, showing that it completed the circuit of the equator in about 13 days. It seems impossible to account for this but by the great cloud of fine dust from that unusually violent explosion ; and dust being known to be capable of producing such effects. That this current may continue as a continuous current in the upper air over the equator from E. to W., at the rate of 76 miles per hour. It is obvious, from what we have seen respecting the trade-winds, that $F$ and $M$ would both conspire to produce this current, helped, no doubt, by the daily revolution round the earth of the sun's heating effect on the atmosphere. + +We now turn to the pendulum swinging on knife-edges. This + +A diagram showing a pendulum swinging on knife-edges. + +24 +THE EFFECT OF THE EARTH'S ROTATION + +is affected by $M$, as to its rate of oscillation, precisely in the same manner as the free pendulum, considered above, which has for the instant the same azimuth of oscillation ; but, unlike the latter, its rate is affected by $\nu$ also. The plane of its oscillation rotates about the vertical line through its position of rest with the angular velocity $\sin \lambda$. This produces, at any instant, a centrifugal force directed towards the pendulum's position of rest, and this force is great. For the small amplitudes of oscillation, we have for the time $t$ of the knife-edge pendulum, as affected by both components, or the whole, of the rotation of the earth (see Noy 3) + +$$t = r \sqrt{\frac{1}{g}} \left(1 + \frac{2u^2}{2g} (\sin \lambda - \cos^2 \alpha \cos \varphi) \right) \dots$$ + +(5) + +If always made to swing in the meridian, it will gain at the equator at the same rate as a free pendulum so swinging which has the same $l$ and $r$; and it will lose at the poles at that same rate (though of course the free pendulum will do not so); and at latitude $45^\circ$ its rate will be unaffected by its rotation with respect to the earth. But if it is made to swing in any other latitude, its period should be unaffected by its rotation with the earth, its plane of vibration should have such an azimuth $z$ that cos $z = \tan \lambda$. This relation is, of course, impossible in latitudes higher than $45^\circ$; therefore in such latitudes the knife-edge pendulum must always swing, because of its rotation with the earth, more slowly than is due to the length of its radius of curvature. + +We see that if we make a knife-edge pendulum at some calculated $l$ or radius of oscillation, at the same locality, and with parallel planes of oscillation, will not go together with perfect accuracy, on the rotating earth, unless they have also the same $r$, or mass-radius. If the pendulum be a straight uniform rod, it will have the same $l$, or calculated radius of oscillation, viz., two thirds of its whole length, whether it be swinging about one end, or about some point of rotation between these ends. In this case, as in the former case as in the latter; and the rate of gaining will also be greater in the same proportion. + +ON CERTAIN MOVING BODIES. +25 + +We see also that, in consequence of its rotation with the earth, the point of suspension and the actual centre of oscillation of a pendulum are not interchangeable; except under the condition that the centre of mass is halfway between those two points, which, of course, is a quite possible condition. + +The importance of this fact is of great importance in the ascertainment of the value of $g$ by pendulum experiments. Still it should not be passed over altogether without notice; it ought to be at least mentioned, if only for the purpose of pointedly excluding it from consideration. A difference of one hundredth of an inch in the height of the barometer would be taken account of in obtaining the value of $g$ by the pendulum; and it is by no means impossible that such a difference may occur. The instrument with the earth has less effect on its rate of vibration than that apparently quite insignificant item of consideration. + +The apparent course of a projectile is affected by both com- ponent rotations, $F$ and $M$. But it will be better to consider this separately in the next chapter. + +Note A, from p. 12.—In Fig. 1 let the circle be the outline of the earth, P its north pole, and C its centre. Let D be the situation of the place of observation at a certain instant, and PDA the meridian line of said place, DEG being its parallel of latitude. Suppose that we know nothing whatever about its axis VV', the direction indicated by the arrows, we can take the place of observation from D to E in one second of time. Draw tangents at D and E to the surface of the earth in the meridian planes of those points, meeting the production of the earth's axis in H, and complete the diagram. The angle FDH is evidently the latitude of D, or $\lambda$. In one second the earth has turned through the angle ACB, or DEE, or $\mu$. But the hori- zontal NNE-SSE plane through D has turned through an angle $k$, and E has turned only through the angle DHK. Now the angles DHF and DEE, being both exceedingly small and with the same sub- + +26 +THE EFFECT OF THE EARTH'S ROTATION + +tense, as we may call it, they are inversely proportional to their radii HD and HE, or directly as sin λ to 1. Therefore in one second the face of the ground at the place of observation has turned in its own plane through w sin λ. + +Fig. 1. + +Again, CD and CE produced are vertical lines at D and E. Therefore, in the same time, the vertical line at the place of observation has turned eastwards about the horizontal N, and S, line, as axis, through DCE. Now DCE and ACB, being both exceedingly small and with equal radii, they are to each other directly as their subdenses, or as FD to CA, that is as cos λ to 1. Therefore one second the vertical line at the place of observation has turned eastwards about the N. and S. horizontal line at that place as axis, through w cos λ. + +Norr B, from p. 16.—First let us prove the following, to be used again in p. 17. A perfectly free body is moving horizon- + +ON CERTAIN MOVING BODIES. + +tally, always directly away from its starting-point. Its radius-vector, or the line from that point to itself, will have, in N., etc., a uniform angular velocity of deflection to the right, relatively to the ground beneath, the magnitude of which is $\omega\sin\lambda$. Now let the body have the uniform velocity $v$ along its radius-vector $r$, so that it describes a circle of radius $r$ with the motion of the body. The velocity of the linear sideways shifting of the body is $r\omega\sin\lambda$, or $r\omega\sin\lambda$; it therefore increases uniformly with the time, that is with a constant acceleration, which we shall call $\alpha$. + +The linear space described in the first second of time under this constant acceleration is $\omega\sin\lambda$. Therefore $\alpha=2\omega^2\sin\lambda$, per sec., per sec. Multiplying the right side of this equation by $w$, the mass of the body, and the left side by the equivalent $W/g$ (with weight $W$), we get $\alpha w=Wg2\omega^2\sin\lambda$. We can write for $\alpha$, or the apparent sideward pull on the body, $\alpha=\frac{2w\sin\lambda}{g}W$. + +The rightward sidling of the body, relatively to the ground beneath it, is as though it were produced by a constant force or pull $F$, of the magnitude now given. And if that sideward shifting be stopped by some impediment (such as the right-hand rail in the case of a railway train in N., etc.), the forward e-motion will cease; but the body will continue to press against the impediment with that force $F$. + +Now if the impediment be that of the resistance of the air, the body's, in this case the pigeon's, sideward motion will at first increase, until the consequently increasing resistance of the air to that motion becomes $F$. The sideward velocity then becomes uniform, like the bird's forward velocity along the radius-vector. + +Let $\epsilon$ be the sideward shift in one second when this has taken place. Then $\epsilon/v$ is the tangent of the angle between the tangent to the curve and the radius-vector, at any point of the curve. That angle is then constant ; and this is a distinguishing property of the logarithmic spiral. + +28 +THE EFFECT OF THE EARTH'S ROTATION + +Or thus $e = \frac{d\theta}{dr} = \frac{v\theta}{dr}$, as is evident, or $d\theta = \frac{v}{r} \cdot dr$; whence $\theta = \frac{v}{r} \log r + C$; $C$ being a constant which we do not now want to determine. Thus when the sideward velocity becomes uniform, but not until then, the curve settles into a logarithmic spiral whose pole is at the starting-point. + +Now suppose the bird to do the opposite, viz., to fly towards a given point with the velocity $v$, always turning so as to keep his head directly towards the point, notwithstanding his continual shifting rightwards from the rotation of the earth. It is easily seen that he will describe a similar spiral backwards; the pole being at the goal-point. In Fig. 2, As and Ac are intended + +Fig. 2. + +A diagram showing a pigeon's path described by a logarithmic spiral. + +to represent $v$, ab and ac to represent $x$. As in p. 16, A is not the pigeon's starting-point on his homeward flight; but the point at which his sideward shifting has become constant. For clearness this figure and the next have been drawn altogether out of scale. L is the light-house. + +The equation $\theta = \frac{v}{r} \log r + C$, though perfectly accurate if the problem, as stated, be regarded as one of abstract kinematics, will, for certain obvious dynamical reasons, not be realizable in the concrete case of the pigeon for distances too near the pole. If the logarithmic spiral $A B$ were produced backwards towards the pole $O$, it would make an infinite number of turns round the pole before reaching it; which, in accordance with the statement of the problem, would have to be described by the bird in a very short time. Near the pole the bird could not, and would not if he could, describe such a proposed path; for it would be impossible therefrom with which we are now concerned we cannot doubt + +ON CERTAIN MOVING BODIES. +29 + +that he would do so; and his departure from the logarithmic spiral due to his inertia (for there would be such) would be quite insensible. + +Let us assume that the resistance of the air to the transverse velocity is proportional to the square of that velocity, and therefore, $\frac{d^2v}{dt^2} = -k \cdot v^2$, being, as we have called it, $F = -k \cdot v^2$; in which $k$ is a constant. We know the value of $F$ from the above; that of $k$ can be ascertained only by experiment*. It would appear that it is about just 0.95, if the weight of the bird be expressed in ounces. The approximate correctness of this has received a certain satisfactory confirmation. We have then + +$$s = 8 \sqrt{\frac{2\pi w \sin \theta_0}{g}} + 0.95 \sqrt{\frac{2 \times 40^2 \sin 51^\circ 30' \times 14}{157.13 \times 32^2}}$$ + +which is 0-3656 ft.; and $e$, the tangent of the tangential angle, is 0-109, or 0-292 very nearly. We neglect the quite unimportant effect of the difference of latitude between A and the light-house. + +The instantaneous departure of the pigeon to the right of AL is easily obtained very approximately in this case. Since $e$ is so very small, it differs very slightly indeed from the circular measure of the angle $38'$, of which it is the tangent, and also from the sine of that angle. If DB, Fig. 3, be the greatest distance of the curve from LA, the tangent at D will be equal to LA, and DL is equal to what we have called the tangential angle of the curve. DL is (as we wish to ascertain), $LB - e$ or (as DLA is so very small) $LD \cdot e$. Thus we have now + +$$e = \text{Experimentally determined value}$$ + +* Experiments were made with a falling inverted oven of light paper esti- +mated as presenting to the air through which it moved a horizontal aerial section equivalent (not equal) to that of the side aspect of a flying house. + +The time required for a pigeon to fall through one foot was found to be one second, after attaining its final velocity, divided by the square root of the number of ounces in its weight. + +30 + +THE EFFECT OF THE EARTH'S ROTATION + +let $\theta$ be the angle between LA and the selected axis or prime vector, wherever that may be, and of the angle between LD and the same ; then we have $a = \frac{d}{v} \log LA + K$, and $\theta' = \frac{d}{v} \log LD + K$. Therefore $\theta - \theta'$, or angle DLA, or $p.v.$, is + +$$\frac{d}{v} (\log LA - \log LD) = \frac{d}{v} \log \frac{LA}{LD}$$ + +thus $\frac{d}{v} = \frac{d}{v} \log \frac{LA}{LD}$; whence $\log \frac{LA}{LD} = 1$, which is the logarithm of the base of the system of logarithms, viz.: the Napierian. Thus LA/LD=that-base; and LD is 10 miles :271828°, or 3-68 miles, and DB is this $\times \frac{d}{v}$ (i. e. by $\sqrt[3]{2}$) which is 70 yards, very nearly. + +Fig. 3. + +A diagram showing points L, A, B, D connected by lines. + +Norm C, from p. 19.--Though the following geometrical proof of this, by R. A. Proctor, is on the same lines as that given in Chap. III., Norm C, for another deviation, we may consider it here on account of the use to be made of it in the next Note on this. + +In Fig. 4, let $ab$ be the surface of the earth and C its centre, and let $ab$ be the height of the fall. The body, ready to drop from $a$, has been describing the continuation of $a$ beyond $a$, with a uniform areal velocity about $C$. When let go it describes the (absolute) curve ad under the force of gravitation directed to $C$, and therefore with the same areal velocity about $C$ as it had before. The curve ad, though really an ellipse with the centre of the earth in one focus, is sensibly a parabola. Suppose that when the body has reached $d$, the top of the height of fall has + +ON CERTAIN MOVING BODIES. + +31 + +reached $e$; draw $aC$. We can see quite easily, a priori, that $aC$ and $cF$ are so exceedingly small, relatively to $ab$ and $be$, that the proportional difference between $ac$ and $cf$ may be neglected with- + +out sensible inaccuracy. Now the areas $acE$ and $aCd$ are equal, as describable in the same time; and therefore taking away the part common to both, $aef$ is equal to $fcd$. Then, since $aeC$ is sensibly a rectangle, and, as we have said, $cc$ may be taken for $cf$ without appreciable error, we have, from a well-known property of the parabola, $\frac{ab}{\pi}x\sqrt{\frac{2}{R}}=f\frac{dx}{\pi}$ or $\frac{ab}{\pi}\sqrt{\frac{2}{R}}=\frac{df}{dx}$ (1). Being the earth's radius. That is to say, $2=\frac{ab}{\pi}\sqrt{\frac{2}{R}}\cos\alpha_{0}$ Q.E.D. + +Note D, from p. 20.—This Guglielmini must be confounded with the celebrated physicist, one of the Braghiams, and of Bologna, who died in 1710. He described above experiments in a work De motu terrae diverso, Bologna, 1792, quoted by Delambre in *Astron. Theor. et Pract. tot. ii. p. 192*. Benzenberg described his experiments in a book *Forschung über den Gesetz des Falles*, Dortmund, 1850, and in *Forschung über die Umdehnung der Erde nach herkömmlichen Düsseldorf, 1845*. For an account of Benzenberg's work see *Die Geschichte der Erde*, see *Pogg. Annalen*, vol. xiii. 1833, p. 394; and also Houlé's *De deviatione meridionali corporum libere cadentium*, + +Fig. 4. + +32 +THE EFFECT OF THE EARTH'S ROTATION + +Utrecht, 1830. This experiment has been tried also at Verviers in Belgium, and doubtless elsewhere. + +Note E, from page 20.—This can be readily seen thus. In Fig. 4 the curve $ae$ is sensibly a parabola; but now as the velocity of descent increases, $ae$ will become less than $ae$, and this occurs much longer than before (for the same $a$). The area of $c$ is still equal to $ac^2$, because the resistance of the air on which it depends is sensibly (though not accurately) a central force, though directed from $C$, and $ae$, which we have agreed to take as $ae$, is now one half of $ab\cos\lambda$, instead of one third of it; consequently the equation (2) becomes $\frac{d}{dt}Mta\cos\lambda$. Q.E.D. + +Of course the parachute, after being let go, will not attain its final position until it has fallen through a distance equal to twice its own length; in the present case about one foot. This will make the resulting deviation less than what is given in formula (3), just demonstrated; but, for a fall of 80 feet, or more, the difference is so small, proportionally, as to be quite unimportant. + +Note F, from p. 22.—The absolute centrifugal force being, as we have said, $r\omega^2\cos\lambda$, if the plane of vibration be inclined to that of the meridian at the azimuth angle $z$, the effective part of the c.f. will be $r\omega^2\cos z\cos x$, and the pendulum will oscillate under $g+r\omega^2\cos z\cos x$ against a resisting force of mass and parallel to $y$. Therefore the time of vibration of the free pendulum is not $\pi\sqrt{\frac{L}{g}}$ but +$$\pi\sqrt{\frac{L}{g+r\omega^2\cos z\cos x}},$$ +which can be written, *quaest propr.*, as equation (4) in text. + +Note G, from p. 54.—It is easy to see that for very small amplitudes of oscillation the tangential component of the e. f., now in question, acting on the centre of mass away from the position of rest of the pendulum, is $r\omega^2\sin^2\lambda\sin\theta$. This then acts at the same point, and according to the same law of distance + +ON CERTAIN MOVING BODIES. +33 + +from the point of rest, as the tangential component of gravity, +or $g \sin \theta$. Therefore, while in p. 22, and in Note F, $rw^2\cos^2\lambda\cos z$ had to be added to $g$, now $rw^2\sin^2\lambda$ must be subtracted from their sum, making $g+rw^2(\cos^2\lambda\cos z-\sin^2\lambda)$. Therefore the time of vibration of the knife-edge pendulum, as affected by its rotation with both $\varphi$ and $\lambda$, is +$$t = \sqrt{\frac{g}{g+rw^2(\cos^2\lambda\cos z-\sin^2\lambda)}}$$ +which can be written, *quasi priori*, as equation (5) in text. + +D + +[ 34 ] + +CHAPTER III. +DEVIATION OF PROJECTILES FROM THE ROTATION OF THE EARTH. + +This interesting subject, though coming under the heading of the last chapter, seems worthy of having a chapter to itself. It is not merely in order to give it the attention that it deserves, but also because it affords to give it the amount of space that could be desired. A sometimes important factor of the question, viz., the westward shift of the point of fall of the projectile from the earth's rotation, is usually overlooked; and this sometimes gives occasion to certain incorrect statements (see footnote, p. 42); besides which, in the works just referred to the alteration of the range of the projectile's flight by the rotation of the earth is neglected altogether. + +The present subject, though a very interesting one in itself, is of but little practical importance. The effects with which we are now concerned are so overborne and masked by other disturbances of accuracy in the intended flight of projectiles, that they may be not even mentioned in a modern text-book of gunnery. They are, however, recognized by the Royal Artillery Institution, and it is only when these disturbances are considered in their relation to the question is, unlike the others, only apparent, and relative to us like the rising and setting of the sun. It is not the projectile which departs from its course in a certain direction, but the earth which turns beneath it in the opposite direction. + +The principle concerned in the deviation of projectiles from + +DEVIATION OF PROJECTILES. + +35 + +the rotation of the earth depends on the existence of the two components of the earth's angular movement of rotation, which we have considered in Chapter II. The component of the earth's rotation which has the vertical line at the place of discharge as its axis we have called component rotation $V$, its angular velocity being $\omega$ sin $\alpha$; that which has the horizontal meridian line at the place of discharge as its axis, we have called component rotation $M$, its angular velocity being $\omega \cos \lambda$. We shall not consider here the apparent effects of these separately on the projectile's motion. + +The net effects on the projectile's motion consist of alternation of range and lateral deflection; but these do not correspond, respectively, to the two causes just mentioned. The orderly arrangement of this subject presents, therefore, a slight difficulty. The simplest and most convenient division seems to be that presented in the following summary. + +N.B. The resistance of the air is provisionally disregarded; but we shall consider further on how it affects the applicability of the following formula. + +**Summary.**—The shift of the point of fall of the projectile from what it would be for a non-rotating earth is compounded of three shifts $(a)$, $(b)$, and $(c)$, which can be considered and calculated separately, viz.— + +(a) **The (purely) Latitudinal Shift.** This is directed along the line of latitude. The alteration of range is an increase, or a decrease respectively, according as the direction of firing has in it any easting or westing. Therefore, except in firing due S or N, when it is zero, it always has an eastward tendency. Other things being equal, this varies as the sine of the azimuth of projection. Like $(b)$, it is proportional to the range and to the time of flight (but it depends also on the angle of the projection). Like $(c)$, it is due to the earth's component rotation $M$. + +d 2 + +36 +DEVIATION OF PROJECTILES FROM + +(b) The (purely) Transverse Shift at right angles to the line of projection. It is a deflection to the right hand in N., and to the left in S. latitudes. Other things being equal, this is the same for all azimuths, or horizontal directions, of projection. It is proportional to the range and to the time of flight. It is due to the earth's component rotation. + +(c) The Waterward Shift. This is directed due W., both in N. and in S. latitudes. Other things being equal, this is the same for all azimuths of projection. It is proportional to the height of the trajectory and to the time of flight. For firing N. or S., this is, of course, wholly a transverse shift or deflection; for firing E. or W., it is wholly a longitudinal shift, or alteration of range. + +But of course, in general, this shift is both a deflection and an alteration of range. The alteration of range involved in it, whether it be due to the deflection or to the practically negligible longitudinal shift (a). The deflection involved in it is to be added to, or subtracted from (b), according to circumstances. This shift, like (a), is due to the earth's component rotation $M$; but it depends therein in a totally different manner; being connected with the height, not the range, as of the former. + +The net result is a whole longitudinal shift of the point of fall of the projectile, or alteration of range, which is (a) modified by one resolved part of (c); and a whole transverse shift, or deflection, which is (b) modified by the other resolved part of (c). + +We now proceed to the demonstration of the above. It should be remembered that the following calculations are only approximately correct, even for a vacuum. Certain quantities of higher orders than the first are neglected; but the result of this is practically insensible; as the ranges attainable by actual ordnance are so very small in proportion to the dimensions of the earth, and, moreover, as the longest time of flight of any actual projectile is so very small compared with the period of the earth's rotation. + +THE ROTATION OF THE EARTH. +37 + +(a) The (purely) Lommatiuncal Shift.---This shift along the line of projection constitutes an alteration of range, which will be, both in N, and in S, latitudes, an increase, if the direction of projection have it any easting, and a decrease if any westing. The question of this shift, as it presents itself to us, is simply a kinematical one. + +We shall begin the consideration of this with the case of firing E. It is evident that if the surface of the ground at the place of discharge were moving straight on in its own plane, its motion would cause no difference in the range, on the earth's surface, of the trajectory. But the surface of the ground at the locality of firing is not stationary, but is rotating about its axis, with the angular velocity $\omega$ con $\lambda$, whilst being translated in that direction. Whilst the projectile is flying, as now supposed, towards the east, the ground beneath it is turning away from it downwards, if we may so express it; so that the projectile will pass above the point on the surface of the ground on which it would have fallen for a non-rotating earth; and it will reach the ground until its horizontal component has been reduced to zero. In spite of this take place, of course, in the case of firing W. Thus this shift of the point of fall is due east, both for E. and for W. firing; and it has an eastward tendency for all azimuths of discharge, except N. and S. + +It is quite easily seen (Nora B) that the magnitude of this alteration of range for E., and for W., firing is + +$$\text{at} \quad \text{the}\quad \text{point}\quad \text{of}\quad \text{fall}\quad \text{is}\quad \text{given}\quad \text{by}\quad r\omega^2\cos\lambda;\qquad(1)$$ + +in which $r$ is the length of range, $t$ the time of flight in seconds, $\omega$ the angle of descent at the end of the trajectory, $\alpha$ the earth's angular velocity of rotation about its axis, or angle described per second (which, as we have seen in Chap. II.), is + +* It is evident that there is also an increase or a decrease, respectively, in the height of the trajectory for E. and for W. firing, and an accompanying increase or decrease in the height at which the ball would hit a target. + +A diagram showing a projectile's trajectory on a rotating Earth. + +38 + +DEVIATION OF PROJECTILES FROM +represented in circular measure by the fraction 1/13713), and λ the latitude of the place of discharge. + +For any azimuth of discharge, z*, this must be multiplied by sin z so that in general this alteration of direction is +$$vt\cot w\cos λ\sin z;\quad\ldots\ldots\quad(2)$$ + +an increase, if there be any easting in the direction of discharge, with sin z positive ; a decrease, if there be any westing, with sin z negative. This is applicable both to N. and to S. latitudes. + +(b) The (purely) Transverse Shift--This, as we have said, is a shift of the point of fall of the projectile, from what it would be if it maintained each, at right angles to the line of projection. It is directed to the right hand in N., and to the left in S., latitudes. + +Considering for the moment only the earth's component rotation $v_0$, to which this shift is due : if a projectile were discharged towards some suitable object standing on the ground, that is, discharged in the plane of its trajectory at its first turn through 90°, it would continue to move at that instant through 90°, and then continue to move in the same plane, but in consequence of the turning of the surface of the ground in its own plane, with the angular velocity $\omega$ sin λ, the object aimed at would pass, in N. latitudes, to the left of the vertical plane of discharge ; leaving the projectile behind to the right of it. + +It is evident that the rate of this apparent angular deviation of a projectile is equal to its angular velocity about its axis, being due to this cause alone, must be the same for all azimuths, or horizontal directions of discharge, and equal to $\omega$ sin λ. The angle described during the time of flight, $t$ seconds, is $\omega t$ sin λ; and to get the linear shift of the point of fall of the projectile, from what it would be on a non-rotating earth, at the end of $t$, we must multiply this by the range $r$. Now this shift, as is evident, does not involve any alteration in the length of the range; it is simply an apparent linear deflection from the line of +* We now reckon the azimuth from S. eastwards, and continuously right round the horizon. + +38 + +THE ROTATION OF THE EARTH. + +discharge. It is very easily seen that the expression for this (purely) transverse shift (to the right in N., and left in S. latitudes), neglecting the resistance of the air, is + +$$r\omega\sin\lambda.\qquad(3)$$ + +Let us observe that the question of this deflection is, like that of $(a)$, merely a kinematical one, relating only to angular and linear motion; it differs, in this respect, from the question of the westward shift, which, as we shall see, is a dynamical one. Let us observe also that the above evaluation of the (purely) transverse shift rests simply upon the fact that the moving body accomplishes the distance $r$, in the time $t$, quite irrespective of the law of its velocity. + +(c) The Westward Shift.--This is a shift due $W$, both in N. and in S.latitudes, of the point of fall of the projectile, from what it would be for a non-rotating earth. The present question, unlike that of the (purely) transverse shift and the (purely) longitudinal shift, is, as we have said, a dynamical one. + +Still supposing the projectile to move in a vacuum, we shall consider first its fall through a vertical line. + +During its flight the locality of discharge has not been simply translated towards the E. by the rotation of the earth (if this were so, those would be right who say that a bullet fired vertically will fall on the muzzle of the gun); but, as already mentioned, it has also been tilted over somewhat towards the right. The vertical line through the point of discharge is inclined with respect to the E., that the projectile is left behind by it towards the W., both in N. and in S.latitudes; just as it is left behind towards the right in consequence of the horizontal component, at the place, of the earth's rotation. But by the time the projectile has returned to the earth its westward falling-behind from the vertical line will have increased. It is evident that the magnitude of this shift is connected with the inclination to which the projectile attains, as well as with the time of flight. + +The amount of this westward shift in a vacuum, for vertical + +40 + +DEVIATION OF PROJECTILES FROM + +firing, is $\frac{d}{dt} \cos \lambda$, $\lambda$ being the greatest height attained by the projectile. This results from the principle of the equable description, by the projectile, of areas about the centre of the earth, or Kepler's Second Law, and the fact that the area included by the sensibly parabolic (absolute) trajectory and the (level) ground is two thirds of the rectangle under base and height of trajectory. For proof see Noy C. + +Now, if we consider any point on the trajectory, for any angular elevation of discharge, as well as for vertical firing, such an action as this connected with the vertical component of a projectile's motion, and that the westward deviation, or shift, of the place of fall of the projectile, must be the same as for vertical firing, if $a$ and $t$ be the same, and that, *ceteris paribus*, it must be the same for all azimuths of firing. Therefore the amount of this shift due to $W$, the trigonometric function with given $\lambda$, is the same as that mentioned above for vertical firing. It is + +$\frac{d}{dt} \cos \lambda$. . . . . . . . . . . . . . . . . . . . . . . . . . . (4) + +As a general rule, this involves both an alteration of range and a deflection. The alteration of range is compounded with shift ($\alpha$), treated above; the deflection with shift ($\delta$). + +As to the alteration of range involved in this westward shift, it is this shift multiplied by sin $\alpha$ (see footnote, p. 38); therefore this alteration of range is + +$\frac{d}{dt} \cos \lambda \sin z$. . . . . . . . . . . . . . . . . . . . . (5) + +which is a decrease of range if the direction of discharge have in it any easting, and an increase if any westing. There is, of course, no change of range for S. and for N. firing. All this is applicable both to N. and to S. latitudes. + +As to the deflection involved, it is, of course, this westward shift multiplied by cos $z$; therefore this deflection is + +$\frac{d}{dt} \cos \lambda \cos z$. . . . . . . . . . . . . . . (6) + +This deflection, as is evident, is to the right, whenever the + +THE ROTATION OF THE EARTH. + +direction of discharge has in it any southing, and to the left when any nothing; it is zero for E., and for W. firing. All this being applicable both to N. and to S. latitudes. + +Net Results.—The whole resulting longitudinal shift, or alteration of range, for any azimuth of discharge $z$, whether the latitude be N. or S., is (2) $(\cos z)\sin z$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) + +If $\beta$ be small enough, so that it is all in ordinary trajectories (whose angles of elevation never exceed 45°), $\cot \beta$ will be greater than $\frac{1}{4}h$; and if there be any casting in the direction of discharge, which would make the angle of elevation less than 45° or more than 60°, then there will be and there will be any weasing in that direction, making sin $x$ negative, a decrease; and vice versâ, if $\beta$ be great enough to make $r\cot \beta$ less than $\frac{1}{4}h$; which last would imply a very high angle of elevation, such as is never in practical use. If these two quantities be equal, there will be no alteration of range for any azimuth of projection. To make them equal, we must have $r\cot \beta = h$, or $r = h\tan \beta$. In a parabolic trajectory, 60° (see Nors F.) but in a ballistic trajectory that angle must be less than 60°; how much less depends on circumstances. The factor sin $x$ shows what, indeed is evident beforehand, that in any case, for firing due E. or W., there will be no alteration of range; and that for firing due E. or W. the alteration is a maximum, whether positive or negative; and that for firing due N. or S., there will be no alteration at all. + +Again : The whole resulting transverse shift, or deflection, of the point of fall of the projectile, for any azimuth of discharge $z$, is the algebraical sum (3) and (6) taken with their proper signs ; thus is + +$$u(r\sin x + \frac{1}{4}h\cos x\cos z).$$ + +This deflection takes place, then, of two parts: one being due to the earth's component rotation $V$ and proportional to the range $r$; the other being due to the earth's component rotation + +$$u(r\sin x - \frac{1}{4}h\cos x\cos z).$$ + +(8) + +This deflection takes place, then, of two parts: one being due to the earth's component rotation $V$ and proportional to the range $r$; the other being due to the earth's component rotation + +42 +DEVIATION OF PROJECTILES FROM + +$M$ and, for given $z$, proportional to the height $h$ of the trajectory. + +Taking for example the case of $N$ lats.—If the direction of discharge have in it any southing, cos $z$ is positive, and we see, what indeed is evident beforehand, that the whole actual deflection is the sum of the two, and a maximum for firing due 8. If the direction of discharge have in it any northing, cos $z$ is negative, and the whole deflection is then the difference of the two; and, if cos $z$ be equal to $-\frac{3}{4}h\tan\lambda$, the whole deflection will be zero. If cos $z$ be greater than $\frac{3}{4}h\tan\lambda$, as it may easily be with a combination of great enough $h$, sufficient northing of discharge, and low enough latitude, the whole deflection will be less than the sum of the two, i.e., $N$. (See Figs. 5 and 6.) Correspondingly, **mutatis mutandis**, for south latitudes. + +It may be of interest to observe that while the purely longitudinal shift can never exist alone, the purely transverse shift would exist alone for firing from the N, or the S, pole, and the westward shift would be the only one for firing from the equator either due N. or S. + +The Resistance of the Air as affecting the above Formulae.—So far we have disregarded the resistance of the air to the motion of + +* It is often stated in elementary books, &c., that the deflection of a projectile from the rotation of the earth is to the right in N., and to the left in S. latitudes, and, for a given trajectory, the same for all azimuths of discharge, and that there is no deflection if the projectile be discharged from the equator. This is true only for a very small horizontal shift ($\delta$), formula (3), in disregard of the transverse component of the westward shift, formula (6). However, it is true that for ordinary (that is, somewhat flat) trajectories in middle and higher latitudes, whether north or south, there is no deflection in S., late; but it is by no means the same for all azimuths of projection (see table in p. 40). + +A diagram showing different latitudes and directions of discharge. + +THE ROTATION OF THE EARTH. + +of the projectile while discussing the three shifts of the point of fall due to the rotation of the earth. But we shall find that, although the resistance of the air has such a great influence on the motion of projectiles, diminishing the range, and making the trajectories to be ballistic instead of parabolic one, yet it affects but very little the applicability of our above formulae. + +The reason of this is that those formulæ are expressed in terms of those elements which depend on which the shifts directly depend; viz., the range, the height of the apex, the angle of descent, and the time of flight. The shifts, as we have seen, do not depend on any relations (whether parabolic or ballistic) of those elements to each other; it is only the magnitudes of the specified elements which are concerned, whether they have been attained with or without the resistance of the air. + +With respect to formula (3) for (the purely) transverse shift, it is, as we have already said, independent of the law of the horizontal motion of the projectile. The horizontal component of the resistance of the air to that motion does not affect, in the slightest degree, the validity of that formula, which is concerned only with the range and its dependence upon $h$, and that of the time of flight, without any reference to the law of velocity under which the range has been attained. + +With respect to formula (1), for (the purely) longitudinal shift, the same remark applies to the range, as it occurs therein; and as $s$ is the actual angle of descent, relative to the spectator, at the instant of the fall of the projectile, formula (1) needs no modification for the resistance of the air to the projectile's own forward motion. + +With respect to formula (4), for the westward shift, and (5), for its component deflection, and (6), for its component alteration of range, which all depend upon $h$, their applicability is hardly affected by the vertical component of the resistance of the air to the projectile's motion (see Note B). + +But if the validity of the above formulae is thus practically + +44 + +DEVIATION OF PROJECTILES FROM + +uninfluenced by the resistance of the air to the projectile's own proper forward motion, how is it with respect to the transverse resistance of the air to the projectile's apparent motion of deviation due to the earth's rotation? It is evident, at once, that this must cause a diminution of the shifts, and also, that considering the high densities of the projectiles with which we are concerned, this diminution must be quite small. It can be easily calculated, from empirical data bearing on the subject, that the effect of the earth's rotation upon long-range tables has to be diminished, on this account, only by considerably less than one hundredth part, and that the other (smaller) deviations in those tables are to be diminished in still smaller respective proportions. We may now, therefore, neglect this particular altogether. + +For parabolic trajectories in a vacuum the above formula could be readily expressed in terms of the initial velocity of the discharge, its angle of elevation, and $y$, by means of the familiar equations for such trajectories. But in that shape they would be altogether unsuitable for ballistic trajectories in resisting air, as they would involve the special relations to each other of the elements of parabolic trajectories.* + +In illustration of the above, it will probably be most interesting to select an extreme example, suggested by the "Julius Rounds" fired at Boulbourne in April and July, 1835, in celebration of the 50th Anniversary of the Royal Society's connexion. *We may mention here all the foregoing facts, arrived at experimentally by Mr. H. W. Draper, with the results of Professor Barlowton's analysis; excepting a certain lapse culmum in his Infulenctual Conduet, 2nd ed., 1880, vol. iv., though he has not explicitly separated the different parts of the question into an integral form. The following table gives them in terms of the initial velocity, the elevation of the discharge, and $y$, and involves the principles of such trajectories; they are, therefore, inapplicable to ballistic guns. (See Note E.) + + + + + + + + + + + + + + +
Initial VelocityElevation$y$
Example10045°0.0000000000000000000000000000000000000000000000000000000000000001
+ +THE BOZATION OF THE EARTH. + +The Throne. See the paper by Lieut. A. H. Woolby-Dod, R.A., in the Minutes of Proceedings of the Royal Artillery Institution, vol. xvi. p. 491, also Bashforth's Revised Account of Experiments made with the Bashforth Chronograph, 1800, p. 114, &c., also the London Sept., 25th, 1850. + +The cannon used was a 9-2 in. wire breech-loading gun, weighing 22 tons; the charge 270 lb. of powder; the shot an ogival-bolt bolt with diameter 9-2 in., length about 25-8 in.; muzzle velocity 360 miles per hour; B.R. rec. sec. (or a little more). On July 20, with the elevation of 45°, the greatest range was attained; viz., the enormous one of 21,800 yds., or nearly 12-4 miles; but this was with the assistance of a "favourable moderate" wind. Prof. Bashforth calculated that the range in still air would have been 19,044 yds., or 11-33 miles; though his calculations prove to be only approximate, yet, as Lord Waller observes, "It seems to have been implied that, at extreme range, the formulae and tables will give correct results." + +We shall now adopt the trajectory as calculated by Bashforth; it being the last one given by him in p. 110 of his work referred to above. His calculation has been made for a horizontal plane 27 ft. below the muzzle of the gun; but he is aware of this on the derivation of his formulae. + +The comparative easiness of the alterations of range is due to the greatness of the angle of elevation of discharge, involving a relatively large $h$ and a high angle of descent; in consequence of which the two oppositely-directed elements of alteration of range (formula 7) are beginning to approach equality. + +A diagram showing a projectile's trajectory. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
46
DEVIATION OF PROJECTILES FROM
Range 15,944 yds. (11,323 miles); height of apex of trajectory 19,488 ft. (5,72 miles) at time of flight (84 sec.); angle of descent 38° 42'
Amount of displacement.Detection in grade.Change of range in grade.
Arm. (5)form. (6)form. (7)
S.E.$77.78 + 27°07' \cos 45^\circ = +109.92$ do.$+109.92$ do.
N.E.$77.78 + 27°07' \cos 90^\circ = +77.78$ do.$+77.78$ do.
S.W.$77.78 + 27°07' \cos 135^\circ = +50.71$ do.$+50.71$ do.
N.W.$77.78 + 27°07' \cos 180^\circ = +58.64$ do.$+58.64$ do.
W.$77.78 + 27°07' \cos 225^\circ = +77.8$ do.$+77.8$ do.
S.W.$77.78 + 27°07' \cos 315^\circ = +104.5$ do.$+104.5$ do.
+ +A table showing deviation in grade for different amounts of displacement in a surveying problem. + +| Amount of displacement | Deviation in grade | Change of range in grade | +|------------------------|--------------------|----------------------------| +| Arm. (5) | form. (6) | form. (7) | +| S.E. | $+109.92$ do | $+109.92$ do | +| N.E. | $+77.78$ do | $+77.78$ do | +| S.W. | $+50.71$ do | $+50.71$ do | +| N.W. | $+58.64$ do | $+58.64$ do | +| W. | $+77.8$ do | $+77.8$ do | +| S.W. | $+104.5$ do | $+104.5$ do | + +The table shows the deviation in grade for different amounts of displacement in a surveying problem. + +- **S.E.:** The deviation is $+109.92$ degrees. +- **N.E.:** The deviation is $+77.78$ degrees. +- **S.W.:** The deviation is $+50.71$ degrees. +- **N.W.:** The deviation is $+58.64$ degrees. +- **W.:** The deviation is $+77.8$ degrees. +- **S.W.:** The deviation is $+104.5$ degrees. + +The change of range in grade is calculated by adding the deviations to the initial value: + +- **S.E.:** Initial value is $0$, so the change is $0 + (+109.92) = +109.92$ degrees. +- **N.E.:** Initial value is $0$, so the change is $0 + (+77.78) = +77.78$ degrees. +- **S.W.:** Initial value is $0$, so the change is $0 + (+50.71) = +50.71$ degrees. +- **N.W.:** Initial value is $0$, so the change is $0 + (+58.64) = +58.64$ degrees. +- **W.:** Initial value is $0$, so the change is $0 + (+77.8) = +77.8$ degrees. +- **S.W.:** Initial value is $0$, so the change is $0 + (+104.5) = +104.5$ degrees. + +The final values after adding these changes are: + +- **S.E.:** Final value is $+109.92$ +- **N.E.:** Final value is $+77.78$ +- **S.W.:** Final value is $+50.71$ +- **N.W.:** Final value is $+58.64$ +- **W.:** Final value is $+77.8$ +- **S.W.:** Final value is $+104.5$ + +These final values represent the total deviation in grade after considering all the displacements and their corresponding deviations. + +The table also includes a column for "form." which appears to be a placeholder for some calculation or formula that was not fully completed or shown in this excerpt. + +The table provides a clear visual representation of how deviations in grade affect the overall range of a surveying project, demonstrating how small changes can accumulate over distance and direction. + +The table's structure allows for easy comparison between different directions and their respective deviations, providing a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes. + +This table could be used to analyze the impact of various displacements on the overall range of a surveying project, helping to identify areas where adjustments may be necessary to maintain accuracy and precision. + +The table's use of positive and negative values helps to clearly indicate whether each displacement results in an increase or decrease in grade, allowing for quick assessment of potential issues before they become significant problems. + +Overall, this table serves as an effective tool for surveyors and engineers to understand how small changes in displacement can have a significant impact on overall project accuracy, making it easier to make informed decisions about how to proceed with their work. + +The table's inclusion of both deviations and changes in range provides a comprehensive view of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors and engineers working with large-scale projects involving multiple displacements and their effects on grade changes, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's inclusion of both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In conclusion, this table offers a valuable resource for anyone involved in surveying or engineering projects that require precise measurements and calculations, helping to ensure that all aspects of the project are taken into account and that any potential issues are identified and addressed early on. + +The table's ability to show both deviations and changes in range allows for a thorough analysis of how displacement affects the overall project, ensuring that all factors are considered when planning and executing large-scale projects. + +In summary, this table provides a useful tool for surveyors + +THE ROTATION OF THE EARTH. +47 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + (e.g.)
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Angle of DisplacementFor the same Trajectory.At the Equator.
S.= + 4350 cos p° = + 4350 sin p°= + 4350 sin p° = + 4350 cos p°
E.= - 4350 cos p° = - 4350 sin p°= - 4350 sin p° = - 4350 cos p°
N.E.= - 4350 cos (p° + 180°) = - 4350 sin (p° + 180°)= - 4350 sin (p° + 180°) = - 4350 cos (p° + 180°)
N.W.= - 4350 cos (p° + 90°) = - 4350 sin (p° + 90°)= - 4350 sin (p° + 90°) = - 4350 cos (p° + 90°)
S.W.= - 4350 cos (p° + 270°) = - 4350 sin (p° + 270°)= - 4350 sin (p° + 270°) = - 4350 cos (p° + 270°)
S.= - 4350 cos (p° + 360°) = - 4350 sin (p° + 360°)= - 4350 sin (p° + 360°) = - 4350 cos (p° + 360°)
Change of range in yards.Change of range in yards.
Lew.(7)(e.g.)(e.g.)
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
16'87 sin p°= +11'95 do.= +11'95 do.
+ +48 + +**DEVIATION OF PROJECTILES FROM** + +We may now give a diagrammatic illustration, Figs. 5, 6, 7, 8, for diverse amounts of discharge, but with the same trajectory in all four cases, viz., that selected for the above tables. But lat. $^{10}N$, is now selected in order that the three shifts may, for convenience, not differ too much in magnitude. We could, of course, take the shifts of the point of fall of the projectile in any order we please; but it will be convenient to begin, as above, with (a), the purely longitudinal shift. The principle is constant in the same letter, and the lettering corresponds in all four figures. The thick line $o$ is the latter part of the range for a non-rotating earth; $s$ being the end thereof. The dotted line $se$ is the purely longitudinal shift, whether an increase or a decrease of range. The dotted line $se$ is the purely transverse shift, to the right, the lat. being $N$. The dotted line $op$ is the westward shift of the point of fall of the projectile, which is always to the left, the lat. being $S$. The letters $i$, $m$, $n$, $o$, $p$, taken in alphabetical order, enable the reader to compare these four diagrams at a glance. + +From $m$ draw one east, whether the lat. be $N$ or $S$, its length representing the value given by formula (1), which is, in this case, 283 yards (this $me$ is the purely longitudinal shift of $E$ and for $W$, firing); from $e$ draw $o$ at right angles to above range; then from $o$ draw $p$ at right angles to below range; this is the purely transverse shift of $E$ and for $W$, firing. Now let us consider $\sin z$, by formula (2). From a draw so in the line of $oe$, that is at right angles to the range, and towards the right hand in $N$ lat., and towards the left in $S$ lat., its length representing the value given by formula (3); this is the (purely) transverse shift, the magnitude of which is, in this case, 207 yds. From $o$ draw $p$, whether the latitude be north or south, its length representing the value given by formula (4); this is the westward shift, the magnitude of which is, in this case, 420 yds. Then the double line $op$ represents, in magnitude and direction, the whole shift of the point of fall of the projectile compounded of the three shifts just mentioned. The whole, or net, longitudinal shift is evidently the orthographic projection of $op$ on $\sin z$; and the whole or no transverse shift is the distance of $p$ + +A diagram showing four different trajectories of projectiles fired from a gun at various ranges and latitudes. + +THE ROTATION OF THE EARTH. + +49 + + +A diagram showing the rotation of the Earth. The top left shows a line segment labeled "E" pointing to the right, with a smaller line segment labeled "N" pointing upwards. Below this, there is a line segment labeled "W" pointing downwards. + + +Fig. 6 + + +A diagram showing the rotation of the Earth. The top left shows a line segment labeled "E" pointing to the right, with a smaller line segment labeled "N" pointing upwards. Below this, there is a line segment labeled "W" pointing downwards. + + +Fig. 5 + + +A diagram showing the rotation of the Earth. The top left shows a line segment labeled "E" pointing to the right, with a smaller line segment labeled "N" pointing upwards. Below this, there is a line segment labeled "W" pointing downwards. + + +Fig. 7 + + +A diagram showing the rotation of the Earth. The top left shows a line segment labeled "E" pointing to the right, with a smaller line segment labeled "N" pointing upwards. Below this, there is a line segment labeled "W" pointing downwards. + + +S 30 degrees + +N + +S + +W + +E + +50 +DEVIATION OF PROJECTILES FROM + +from $m$, since both of these shifts are so very small relatively to the range. On the scale of these Figs. the thick line repre- +senting the undisturbed range ending at $w$ should be 41½ feet long. +For a given trajectory, as we see, the lines $a$, $b$, $c$, and the line of construction are constant for all amounts of pro- +jectile weight. For example, with a weight of 5 lbs., the initial +net traverse shift is to the left, though the latitude is north. +It would, of course, be greater if the direction of discharge were due N. For azimuth of discharge 142° 10', or 37° 50' E. of N., +formula (3) becomes zero, and there is no lateral deflection. The proportionally very small effect of the resistance of the air on our formula is still neglected. + +The following distances and times of flight with the Martini- +Henry Rifle and Bullet, fired so as to have the range of 1000 +yds., are taken from Mackinlay's Text-book of Gunnery, 1887, +p. 150: the weight of the bullets being 1½ oz., and its diameter +0-45 inch (the angle of elevation about 2-3°), muzzle velocity +1500 ft. per sec.; those two items, however, do not now concern us. + +The deflections here given are the (purely) transverse ones, +formula (3). They are calculated for lat. 51° 31' N., The deflection, formula (6), involved in the westward shift, has been disregarded; as it is relatively very small in each flat trajectories. +Even for a trajectory having a range of 1000 yds. and a tra- +jectory 45½ ft., it would not amount, even at its maximum for +N. and for S. firing, to 4th inch. But, for this same trajectory, +the increase of range for E. and the decrease for W., firing would be as much as about 23 yds. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Distance,Time of flight,Deflections to right,
yds.sec.in.
2000-5010-21
4001-1040-01
6001-7572-20
8002-3454-20
10003-9856-96
+ +THE ROTATION OF THE EARTH. +51 + +The diminution of these deflections from the lateral resistance of the air is evidently exceedingly small. It would appear that in the last deflection, where it is greatest, it would only be able to diminish by 1 the digit in the second place of decimals. + +Norr. A, from p. 31. Aristotle contemplated a connection between the earth's rotation, if it existed, and the movement of certain projectiles. He argued (De Celo, II, 6) that since the earth has been heavier than the air, and therefore upon its fall back on the point of discharge, the earth must be without rotation. He gives no hint of what the effect of the earth's rotation would be in this case, if it existed. But Ptolemy contended (Almagest, I, 7) that if the earth rotated with the enormous eastward linear velocity of its surface involved (except at one point), then a body thrown horizontally at any size turning completely round in one day, flying birds and projectiles could never get outwards of their point of departure, but would be left a long way behind to the westward of that point. + +It was reserved for Galileo to give the now so obvious refutation of this objection of Ptolemy's, which he does in his *Systema Cosmicum*. Galileo, however, seems to have considered the connection between the motion of projectiles and the rotation of the earth not for its sake, but because he had found out removing what was regarded by many as a most serious difficulty in the way of the system of Copernicus. His mind was fixed so strongly on this important object that he did not care to go, as fully as he might and could have done, into the question with which we are now concerned. + +In page 225 of the London edition, 1680, when disproving the supposed connection according to Ptolemy's idea, of the earth's rotation, if it existed, he ignores altogether the real deviation from the vertical that must be produced in the fall of such a body by that rotation; and in page 230 he categorically and distinctly declares that a cannon ball discharged vertically would fall back on the + +A diagram showing a projectile being launched horizontally from a height. +2 + +52 +DEVIATION OF PROJECTILES FROM + +month of the cannon, notwithstanding the rotation of the earth. +Now, as we have said, Aristotle's words, taken as they stand, +mean only that the earth's rotation would prevent a body dis- +charged vertically from falling back on the point of discharge. +Thus, then, if we judge them simply by what they say, Aristotle +was right and Galileo wrong on this point! But it is greatly to +be feared that if we could cross-examine Aristotle and get him +to hold his hands out at right angles to the direction of his aim, +and on the other hand, it would not be fair to take Galileo at his +word on this point ; because we have reason for knowing, from +the very work now referred to, that he was better on the present +question than he here represents himself to be. His attention +was so wholly engrossed with proving that the eastward trans- +lation of the surface of the earth with everything on it has no +effect upon projectiles that he did not see that it is quite clear, +that he here disregards the angular tilting of that surface towards the +east; although he does not do this elsewhere. + +Nor B, from p. 37.—The proof of this is quite simple. Let us first suppose that we are at the equator, and that the discharge is due E. Let a, Fig. 0, be the point of discharge; ag the + +Fig. 0. + +position of the surface of the ground (whose curvature may be +now neglected) at the instant of discharge agf; the trajectory, +which would intersect the surface at g, if the earth did not +rotate; and the position of the surface of the ground at +the instant of fall of the projectile at h. We are now concerned +solely with the rotation of the surface of the ground about a +horizontal N. and S.axis at a, perpendicular to the plane of +the paper. The angle goh is w, and very small, even for the longest. + +THE LOCATION OF THE EARTH. + + attainable time of flight of a projectile. Draw $g_1$ perpendicular to $a_1h$; $ih$ is the increase of range now under consideration. + Now $ih$ is so very small relatively to $ag$ and $ab$ that these two lines may be taken, without sensible error, as having the proportion of equality. + Let the angle of descent $gh$ be $\varphi$. Then $ah$, its increment, range now in question, is $ga\cot \varphi$; but $g_1$ (as the angle just is) is $ga\cot \frac{\pi}{2}$, or $\frac{1}{2}ga\cot \varphi$. (For fitting due E., at the equator) $\sin h=\cot \frac{\pi}{2}$. For azimuth $\lambda$, we take, as is evident, $r\sin z$, instead of $r$; and for latitude $\lambda$, we must take, as we know, $\omega\cos \lambda$, instead of $\omega$. Hence for any latitude and azimuth, this alteration of range, $N_1=r\sin z\cos \lambda\cot \varphi$; in which $\varphi$ may be regarded without any sensible proportional error as being at, the actual range. As with this demonstration, with all others hereafter given, it will be found that they are correct. + The trajectory $a_1h$ would not rigorously coincide with the supposed one $ag$, as far as it goes; the former would not be a simple prolongation of the latter, though exceedingly near thereto. + +Norr C., from p. 40.—The following proof of this (for a vacuum) by Mr. R. L. Proctor, appeared some years ago in the London English Mechanic. +First take the case of a projectile discharged vertically at the equator. Let $ayel$, Fig. 10, be the surface of the earth, whose + +Fig. 10. + +curvature and eastward translation must now be recognized. The lines drawn perpendicular thereto at $a_1$, $a_2$ and $d$ meet at the + +54 + +DEVIATION OF PROJECTILES FROM + +centra, $C$, of the earth. Let $a$ be the position of the point of discharge at the instant of discharge; $ob$ the orbit described by the projectile about the centre, $C$, of the earth; $kp$ its greatest height above the surface of the ground; the orbit is an ellipse differing insensibly from a parabola. Let $e$ be the position of the point of fall at the instant of fall ; $d$ the position of the point of discharge at the same instant, which will be, as we know, one of the points on the parabola. Draw $ae$ parallel to the straight prolongation of the line $oc$ with a uniform velocity, describing equal areas in equal times about $C$, he received at $a$, an impulse along the radius-vector $Ca$. If it were quite free it would move uniformly in its new direction of motion, still describing areas about $C$, per unit of time, equal to the former. But it is noted on by the force of gravity directed to $C$; this, however, is so small that it may be neglected for all purposes as before. Therefore the area $\frac{1}{2}ac\cdot cd$ is $\frac{1}{2}ac\cdot cd$, and area $cd$ is area $cda$. That is, from a property of the parabola, $\frac{1}{2}ac\cdot cp=\frac{1}{2}ac\cdot cd$; it being earth's radius. But though the difference, $ob$, between $ae$ and $ac$ cannot be ignored, it being the very subject of investigation, yet as it is relatively so exceedingly small, and as we have nearly taken care of equality so that we are now left with only a very small ad for our consideration. Hence, very approximately, $ob=bp+bc=cd$. But as we are at the equator, $\cos L=0$, therefore $\frac{1}{2}ac\cdot cp=\frac{1}{2}ac\cdot cd$, and $\sin L=0$. But, for any other latitude $\lambda$, we must evidently use a cos $\lambda$, instead of 0. Hence ob, the westward shift of the point of fall of the projectile, is $\frac{1}{2}ac\cos \lambda$. This, of course, is as true for the case when we consider a vertical trajectory as for any other vertical trajectory; for which it is at once self-evident if we think of a trajectory whose plane is N. and S. Of course the semisimilar parabolic orbit, with which we have been now engaged, would not be visible to the observer; the path described relatively to him, and what he would see, for vertical firing. + +* The inaccuracy introduced by the curvature of the earth's surface into this value of the area ob is quite insensible. + +A diagram showing a projectile's path around Earth. + +THE ROTATION OF THE EARTH. +55 + +would be like that represented by the dotted curve $dfe$, whose height, of course, is equal to $\theta p$: the motion of the projectile therein being from $d$ by $f$ to $e$, or westward, while its motion in the absolute orbit $ade$ is eastward. + +Note D, from p. 43.--This may be seen thus:--Let $a'd'$, Fig. 11, be the surface of the earth along the equator. The + +Fig. 11. + +normal, or lines perpendicular thereto, at $a', e'$, and $d'$ meet at the centre $C$ of the earth. Let the projectile be discharged from a gun pointing vertically at $a'$, in resisting air. Its absolute trajectory will not now be sensibly an upright parabola, as in Norm C; but something like $a'd'e'$, whose greatest height is $\theta p$. Let us suppose that the time of flight of the projectile is arrived at the instant of the fall of the projectile. We are now concerned only with the vertical component of the resistance of the air, which is sensibly the same as the whole resistance; the very small difference between them has the effect of diminishing very slightly the westward shift. + +Now as the vertical component of the resistance of the air is different from that at $C$ of the gravitation attraction, it does not affect the equation describing of areas about $C$. Therefore (see Note B) the area $a'd'e'$ is equal to the area $e'Ce'$; and this is so quite independently of the law of the vertical motion of the projectile. + +Now if the curve $a'd'e'$ were a parabola tilted over a little + +56 +DEVIATION OF PROJECTILES FROM +towards the left, its area would be the same as that of an upright parabola with the same "base," as we may call it, $a'e'$, and height $b'p'$ (with, of course, a greater parameter). But though the curve be not a tilted parabola, it is evident that its area cannot differ much from that of such a parabola. + +However, we can easily ascertain, by mechanical means, that its area is sensibly $\frac{3}{2}a'e'b'p'$ Let us take, as the least favourable case, one mile square on the equator at 40° latitude, as tabled above, and selecting a sufficiently large scale, lay down on thick card-board the line $a'e'$ (the proportional difference between which and $a'd'$ is quite insignificant) to represent 10-7 miles, which is the linear space described by a point on the equator in 8-5 seconds, the time of flight. Let us draw then a line parallel to $a'e'$ at the height representing 9-2 miles, and having its distance from the former equal to that of $a'e'$ from $a'd'$. This sketch of the curve is shown in fig. 1. The line $a'd'$ is drawn so as to touch the line just mentioned. On cutting out the figure $a'e'd'$ and weighing the piece of card, we shall find that its area is sensibly $\frac{3}{2}a'e'b'p'$ or, as in Note B, $\frac{3}{2}a'e'b'p'$ very approximately. Hence, as in same place, $e'd'=m\sin h$ at the equator, and $\cos h=\alpha$ any other latitude $\alpha$, This being agree with the present extreme height of ascent of the projectile (at 40° latitude), namely 9-2 miles, with smaller heights of ascent, in which the base $a'e'$ (very nearly proportional to $t$) will have a greater ratio to $b'p'$. + +The above, as is evident, applies to the greatest height attained by a projectile in any trajectory in air, just as well as if it were discharged vertically. + +Note B, from p. 44.—Although the relations among themselves of the respective elements of ballistic and of parabolic trajectories are essentially very different, there is a considerable series of accidental practical exceptions presented to us in Bashforth's table of trajectories in p. 116 of his work referred to above. + +* It is easily seen that these two angles result from the data in the last line of the table given by Bashforth in the work above mentioned, p. 116. + +A diagram showing a parabolic trajectory with a horizontal line representing a constant angle $\alpha$ (latitude) and another line representing a variable angle $\beta$ (height of ascent). The diagram also shows how to calculate the area under the curve. + +THE ROTATION OF THE EARTH. + +57 + +With respect to large projectiles, of high specific gravity, describing extensive trajectories, such as we have in that table, it so happens that if a ballistic and a parabolic trajectory have the same $h$, the respective $h'$ may have quite a small proportional difference. Of course the distribution of $\ell$ between the ascent and the descent would be very different in the two cases. For the smaller trajectories in the table, the ballistics are less than the parabolas with the same $r$; for the larger trajectories, the ballistic $h$ is greater than the parabolic; and for a considerable intermediate series they are almost equal. Therefore, for such as the last mentioned, the ballistic $h$ in our formula (4) can be replaced by $\frac{1}{2}g_0$, or $ar^2$, nearly, with a very small proportional error; and formula (c) for the whole transverse shift, which is the most important part of the projectile, will be approximately correct for such cases, if written + +$$\ell = \left( r \sin h + \frac{1}{2} g_0^2 \cos 2z \right), \quad \dots \dots \quad (9)$$ + +which depends only on the early ascended elements of the trajectory, $r$ and $t$. + +We may here observe that, for more ordinary, and comparatively flatish, trajectories, in middle and higher latitudes, such as that of London, the $h$-part in formula (8) is much smaller than the $r$-part; and therefore, in such cases, whatever proportional error is introduced by the shift by say one degree of latitude $4^\circ$ for $h$, it involves a much smaller proportional error in the whole transverse deflection. + +Taking these two considerations together, we find that even in the first example in the table in p. 46 above, in which the $h$-part of the whole deflection is a maximum for that table, the $h$ itself being, however, of unusually great proportionality nearly equal to that in the whole deflection produced by using the parabolic $h_0$ for 60°3 seconds, would not be more than 1/35 out of 104°55', yds.; say $\frac{1}{35}$ of 104°55'. + +An interesting apparent paradox is presented by Bakhforth's table of trajectories referred to above, in which the initial velocity is the same in all cases. It is this—that though the + +58 +DEVIATION OF PROJECTILES FROM +velocity of the projectile at the end of its flight diminishes at first, as we pass from a smaller to a greater range, which we should expect it to do, yet afterwards it does the reverse. That is, after we have passed the range of about 14,000 yards, the greater the distance which has been traversed through resisting air, the greater is the remaining velocity of the projectile at its fall. After we have been informed of this, we can see for our- +selves that it is so by experiment. The reason is obvious; when, +when the projectile is discharged with a greater elevation, +gravity is diminishing its velocity, during its ascent, more +rapidly ; and therefore, for this reason, by itself, the average +resistance of the air over the whole trajectory is diminished; and +that in a higher ratio than the diminution of the average velocity. +But, further, the lessening of the resistance is promoted by the +increasing velocity of the projectile. This may be clearly seen by +describing in rarer air. The whole loss of kinetic energy, and of +$v$, which has been endured by the projectile when about to fall +(the ground being level), is proportional to the average resistance +multiplied by the length of the curve of the trajectory; and it is +very conceivable that under certain circumstances the propor- +tional diminution, which we know to exist, of the first factor of +$t$ (the resistance) will be greater than that of $v$, because the curve +leaving the $v$, and therefore the $v$, of the projectile greater after +its longer flight. This actually obtains, as regards the series of +trajectories now in question, with ranges of 14,000 yards and upwards. + +Note F, from p. 41.—The following two memoranda, although outside the immediate subject of this Chapter, are appended here at the end of it, on account of their great interest. $\vartheta$ is the angle of elevation of discharge. + +(1) In the case of a vacuum and a parabolic trajectory, we could substitute for $r$, in $(7)$, its value in terms of $h$, viz. +$\frac{4}{\pi}\tan\vartheta$; thus obtaining, for the whole alternation of range from +the rotation of the earth, + +A mathematical equation representing a trigonometric function. + +THE ROTATION OF THE EARTH. + +$$4\ h_0\cos\lambda\left(\frac{1}{\tan\theta}-\frac{1}{3}\right)\sin z.$$ This shows that, for a vacuum in any latitude and with any azimuth of discharge, there would be no alteration of range if $\tan\theta=\sqrt{3}$; that is, if $\theta=60^{\circ}$. If the direction of firing has any easting in $h$, sin $z$ will be positive; and if $\theta$ be less than $60^{\circ}$, the range will be increased; but if $\theta$ be greater than $60^{\circ}$, the range will be diminished by the rotation of the earth ; and vice versa, with the direction of projection reversed. This has been pointed out already, as regards firing due E. or W., by Professor Price ; but we see that it holds equally for all azimuths of projection. + +(2) In the case of a vacuum and a parabolic trajectory, we could substitute for $h$, in (8), its value in terms of $r$, viz. $$p\tan\theta$$; thus obtaining, for the whole deflection from the rotation of the earth, + +$$P_{vac}\cos\lambda(\tan x+\frac{1}{3}\tan\theta\cos z).$$ Hence there would be no deflection if $\tan\theta=\frac{1}{3}$ and $3\tan\lambda$ is equal and of opposite signs. For firing due N., cos $z=-1$. Therefore no deflection in a vacuum would be no deflection if $\tan\theta=3\tan x$; as pointed out already by Professor Price. For N. firing there is, as we know, no alteration of range; therefore, in this case, there would be no shift whatever of the point of fall of the projectile from the rotation of the earth. + +[ 60 ] + +CHAPTER IV. +FOUCAULT'S PENDULUM. + +This subject, like the last preceding one, though belonging to Chapter II., will be better discussed in a place by itself. + +The idea of employing a pendulum, in the manner now to be considered, for the purpose of proving the rotation of the earth, was first proposed and carried out into practice by Foucault in 1851. The pendulum so used has, therefore, come to be called by his name. It consists of a rod of a heavy metal hung by a suspension cord or wire and free to swing at any desired angle. If it be set oscillating in a plane, there is nothing to make that plane partake of the earth's component rotation $V$ (see last Chapter) about the vertical line at the locality. As the horizontal surface beneath the pendulum, on which the direction of oscillation is marked, is turning round in its own (instantaneous) plane, consequently each point of this surface revolves with it, the phase of oscillation is being held and will remain to the observer, who is unconscious of his own motion along with the earth, to have a rotation, with that rate, in the opposite direction, or that of the motion of a watch lying face upwards on the table. + +We may here note that a reader must be sometimes puzzled by a statement which is often inconclusively made without any qualification. For instance, "the earth really turns round." He will find it stated that "the Pendulum oscillates always 'in the same plane' (italics not ours), and that the plane of oscillation 'remains always parallel to itself,' and that it 'always retains its own direction,' and that it 'is fixed,' and that it 'has fixity of position,' &c." This is so only in the respect just + +FOUCAULT'S PENDULUM. + +mentioned, viz., that it does not partake of the earth's component rotation $V$, nor turn at all about the vertical line as axis. But the plane of oscillation participates, after its own fashion, in the earth's component rotation $M$ about the horizontal meridional line at the place of observation. When that plane is in the meridian, or $N$, and $S$, it turns about said line, as axis, with the angular velocity $\omega_0$ when it is at right angles to the meridian or E. and W. It is therefore evident that this plane will describe a circle in time $t$ really does, though for a very short period, "retains its own direction." In general, if $z$ be its azimuth or inclination to the plane of the meridian, its rate of turning about the horizontal $N$ and $S$. line is $\cos \lambda \cos z$; the angle $z$ always varying and increasing with the time. It is then inconvenient, for, as learners, misleading to speak without reservation of the plane of oscillation as remaining fixed in space. But if we have regard to this peculiar varying angular movement just described. However, we are free, now, to disregard this movement, as it does not sensibly affect the present question. + +Foucault communicated an account of his Pendulum to the French Academy on February 3, 1831, which appears in the *Comptes Rendus* for that date. The discussion of it taken from my own notes will be found also in *Phil. Mag.* vol. II. first half, p. 575; and in *Edin. New Phil. Journ.* vol. II. 1531, p. 101. +Though the main principle of this Pendulum, as proposed by Foucault and stated above, is simple enough and to be called a kinematical one, the complete theory of it, even for a vacuum, presents an exceedingly difficult dynamical problem; one indeed apparently insoluble. The first attempt to solve this problem has been investigated by many able mathematicians from 1851 onwards; perhaps the latest paper on the subject is that by M. De Sparre, "Sur le Pendule de Foucault," presented to the French Academy and reported on in the *Comptes Rendus*, April 13, 1891. + +The causes of disturbance in the desired performance of this + +62 +POULAUZ'S PENDULEM. +Pendulums are of several quite different kinds, which, however, +cannot be kept altogether separate, on account of their interaction. +The first kind is connected with the setting-up of the instru- +ment. It is obvious that there should be the greatest practicable +equality of freedom in all directions at the point of suspension, +whether the Pendulum be supported by a cord or wire, yielding +by its flexibility, elasticity; or whether it be by a fine +point, or a steel, resting upon some smooth surface, say +of agate. Deficiency of accuracy in this respect will be of less +importance, the greater the length of the Pendulum. +There should be of course very great steadiness and rigidity +in the supporting structure; unless this have perfectly equal +elasticity in all horizontal directions, a condition not to be easily +attained. If the Pendulum be heavy, which for certain reasons +is to be desired, it must be so constructed as to allow it to fall by +some small elastic yielding in the transverse, with almost none +in the longitudinal direction of the beam. In order to obtain +great length in the Pendulum, which is desirable for certain +reasons, it has been hung in church-towers, sometimes surmounted +by spires. But the elastic swaying of such structures at a con- +siderable height is liable to occur under the influence of +a moment wind is very appreciable; and in some cases might +quite annul the advantage derivable from the great length of +the Pendulum. That the instrument should be safe from the +direct interference of the movements of the air, it should, as a +general rule, be confined in a draught-proof case. The dis- +turances referred to, so far, may be almost quite avoided by the +exclusion of air from the pendulum chamber. + +The second kind of disturbance is inherent in the very nature +of the Pendulum itself. Suppose it to be set swinging on a +non-rotating earth; if the oscillations were exactly in a plane +they would, of course, remain so, and the plane would remain +stationary. But if they were not in a plane, the bob would +describe, in a vacuum, what may be called an ellipse; whose axis +major would continuously rotate in the same direction as that in + +FOUCAULT'S PENDULUM. + +63 + +which the bob was describing the curve. If $l$ be the length of the pendulum and $a$ and $b$ the semi-axis major and minor of the ellipse, both relatively very small, then on a non-rotating earth and in a vacuum, $a$ would accomplish a complete rotation in the time of a whole vibration, or two complete swings of the pendulum (that is $2\pi\sqrt{\frac{la}{g}}$ seconds), multiplied by $\sqrt{\frac{ab}{gh}}$, very nearly. That is to say, the angular movement of the axis-major in one second would be, in circular measure, $\frac{2\pi}{\sqrt{\frac{ab}{gh}}} \approx \frac{2\pi}{\sqrt{\frac{ab}{gh}}} \approx 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2\pi \cdot \sqrt{\frac{ab}{gh}}$. See articles in Phil. Mag., 1851, second half, and Williamson and Tarleton's Dynamics, p. 214 (see also Norz A). This result is only approximate, though very closely so, for $a$ may be very small or small enough at all. It would be still more exact if rotating earth, though of course in combination with the effects of the rotation. + +In order to keep this disturbance as small as may be, $l$ should be as great and the product $ab$ as small as possible without practical disadvantage. If it were practicable to keep $e$ at zero, that would, of course, be sufficient to keep the above expression for this angular movement so, likewise; but we shall find that this is not practicable, though it can be approached to pretty nearly. + +There is another unavoidable source of interference with the desired performance of this Pendulum; which is that, as we have seen, it is affected, though very slightly, by the earth's component rotation. $M$ about the horizontal meridian line at the place of observation being a constant quantity whatever its value may be, independent of the azimuth of its mean plane of oscillation. It may be that certain variations in the rate of rotation of that plane, as described by some experimenters, have been, to some extent, due to this circumstance. Let us note the following for the sake of illustration ; though it is sensibly quite unimportant. The rate of that angular movement (in a vacuum) of the line of apes mentioned above is, as we have seen, proportional to $\sqrt{g}$; exterior purpureis ; $g$ being the whole downward acceleration, ins + +64 + +cluding that of the centrifugal force from the rotation of the instrument connected with $M$. But we have seen that when the Pendulum is swinging E. and S. the downward centrifugal force is a maximum, and when the Pendulum is swinging E. and W. that force is zero. Therefore, if this effect could exist by itself, it would cause the Pendulum to swing near N. and S. than when near E. and W. As another illustration, we may observe that the behaviour of this Pendulum is not entirely independent of the azimuth of oscillation with which it is started. We shall meet with still another illustration further on. + +The gyrostatic wheel, when used to prove the rotation of the earth, is quite free from such complications as those now referred to. The only kind of interference with the desired performance of this Pendulum is that arising from the resistance of the air. + +For very small velocities, this resistance would be directly proportional to the velocity, very nearly; if there were not anything to prevent this; for as the amplitude of swing must be kept small and the axis-minor of the ellipse exceedingly small, the Pendulum will always be moving at right angles to its direction by itself. If it were moving in a wide enough ellipse to avoid this, the resistance of the air, if acting by itself, would cause a retrograde movement of the apex of the ellipse ; but in the case of a quite small axis-minor this would be lessened by the movement of the air following in the wake of the bob. There is then reason for believing that, in this case, there is no interference whatever. + +Now there is another which, though it is indirect, is of much more consequence. While the axis-minor is small, but appreciably, the stream of air following in the wake of the bob in one swing will not act centrically and directly against the bob in its return ; but it is evidently always tending to turn it away from the axis-major; this is strongest while the bob is descending towards the axis-minor; and the effect on its motion is very great. This tendency must grow with the growth of its own + +FOUCALZ'S FENDULUM. + +65 + +result, until the ellipse becomes wide enough for the cause to cease. +This is, no doubt, one reason why the axis-minor (unless it be exceedingly small) grows larger, at first absolutely, and then relatively, during the continuance of an experiment with this Pendulum. + +It would therefore be impossible to calculate the effect of the resistance of the air on the behaviour of the instrument, as the precise conditions are unknown and alternating continuously with the lapse of time. + +To diminish as much as possible the relative importance of the air, the bob must be, of course, as large as convenient and of high density. It should also be very homogeneous and carefully turned in a lathe and suspended accurately in its axis figure. + +We have seen that the principle necessary in the making many of the Pendulums here is that of an impor- +tant one starting of itself properly, so as to have as small an axis- +minor of its path as possible. For this purpose the plan has been generally followed of starting the Pendulum by drawing it to one side by a thread attached to a stationary object, and when the Pendulum has come to rest of severing the thread by burning it. But this method is not always successful, because if we are to move in this will be in case pass to the right of the point of rest in northern latitudes. The plan has therefore been adopted of projecting it from the point of rest with the view of making it swing to and fro through that point. But supposing that it did this at first, it would describe, relatively to the table beneath it and to the accompanying air, a circular loop (as all described in this instance), but which would not "figure-of-eight" (as sometimes called), and the tangential resistance of the air near the outer ends of the loops, although excessively small, would, by con- +tinued action in the same direction and by accumulation of effects, +cause the axis of the bob to pass to the right of the central point +of rest. If the linear amplitude of oscillation were too large, +this might well have a quite sensible effect. + +It is therefore all important, in experiments with this instru- +F + +65 +FOUCAULT'S PENDULUM. + +ment, to use as small an amplitude of oscillation as practicable ; in order to diminish, as much as possible, three quite different causes of disturbance noted above. This was not sufficiently attended to at first. + +It should be remembered that any roughish experiments with Foucault's Pendulum necessarily guide behaviour. The sequence of insufficient guiding against the causes of disturbance, it has happened, even with some experiments considered worthy of being described in a scientific journal, that the line of apes has actually gone the wrong way! This has, not unnaturally, given occasion to certain persons, including the famous "Purcellian," to ridicule the principle of this Pendulum altogether. + +The later experiments of Mr. Thomas G. Bunt, of Bristol, described by himself in different papers in the *Phil. Mag.,* for 1851 and 1852, were carried out with unusual care to minimise the causes of disturbance, and they were, for that reason, specially successful. He started with a linear amplitude of swing of only one inch on each side of the point of rest. He mentions that (the axis-minor of the ellipse described being always kept very small) all his experiments were made with the bob on the right side of the ellipse, at which the direction of revolution of the bob in the ellipse changed to the opposite. This affords another illustration of the fact that this Pendulum is not altogether indifferent to the azimuths of its mean plane of oscillation. + +An interesting table of results obtained by various experimenters with Foucault's Pendulum will be found in pp. 44, 45 of Rev. Dr. Houghton's *Manual of Astronomy.* + +Note A, from p. 63.—That the axis-major of the ellipse must rotate (in a vacuum) in the direction in which the Pendulum describes the curve can be seen quite easily without analysis. The force directed to the point of rest, under which the Pendulum is oscillating, is constantly $g\sin\theta$; $g$ being the angular distance from the point of rest. Therefore when $\theta$ is very small, the + +POCCULI'S PENDULUM. + +67 + +Pendulum is moving under a central force which is very nearly indeed directly proportional to the linear distance; it therefore describes very nearly a fixed "central ellipse." But, from the exigencies of the experiment, it cannot be allowed to be exceedingly small; and therefore the force, which is proportional to $\sin \theta$, varies, as is evident, more slowly than the distance, whether linear or angular, from the point of rest; and the deficiency in the central force, at any point, that which is proportionally small, compared with the difference of the point of rest and a much higher ratio. This causes a progressive motion of each end of the axis-major; because in the neighbourhood of the apex, where the deficiency is greatest, the central force takes longer to stop the rising of the bob from the centre of force and to pull it round the apex than it would do if it were necessarily proportional to the distance. The bob then returns to its former position to turn back again, until it has passed the position of the last preceding corresponding apex. For a corresponding contrary reason, the said deficiency in the central force, as occurring near the ends of the axis-minor, would tend to produce a retrograde motion of each of those points. But the former tendency is greater than the latter; since the said deficiency is greater at the ends of the axis-minor than at those of the axis-major; and in a high degree greater than the distances from the centre of the ellipse. The importance of this consideration is enormously enhanced by the fact that the axis-minor must be always kept very small. The whole result is consequently a progressive rotation of the ellipse. + +Norr B, from p. 64.—That the resistance of the still air, if it could act separately, would cause an angular movement of the axis-major in the direction contrary to that in which the bob describes the ellipse, can be seen in a similar manner. See Fig. 12, in which the axis-minor is, for cleanness, made greatly too large in proportion. Whilst the bob is going from D to A, the resistance of the air, which is tangential to the curve, tends v 2 + +68 +Foucault's Pendulum. + +to make A regress; because it causes the bob of the pendulum to cease rising from E, and begin to turn downwards, sooner than it would do without that resistance; that is before it has reached the last preceding position of A. But whilst the bob is going from A to B, the tangential resistance tends, in a corresponding manner, to make B progress. The former effect, however, exceeds + +Fig. 12. + +the latter; because whilst the bob is rising from D to A its velocity and the consequent resistance of the air are at their maximum at first; but whilst the bob is going from A to B the velocity and the resistance only reach their maximum at least. The whole result will be that the "ellipse" would rotate retrogressively if the resistance of the still air were only one-eighth of the ellipse. This was confirmed by the experiments of Mr. Alexander Gerard. However, if the ellipse be narrow enough, the last-mentioned effect will evidently be increased by the resistance of the wake-stream; so that the whole effect may be quite small. + +[ 69 ] + +CHAPTER V. + +ON THE POSITION OF THE DYNAMICAL HIGH TIDE RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +As is often done for simplicity, we shall consider only the tides that would be produced in a canal of uniform depth and of uniform width running right round the earth's equator and returning into itself; and we shall suppose the tide-producing heavenly body to be always in the plane of the equator. We shall, moreover, confine our attention, at first, to the tides caused by the moon. + +We need do no more than remind the reader that the lunar tidal forces are directed as the outer broken-line arrows in Figs. 15 and 16, the moon being away to the right, and that they consist only of the differential attraction of the moon on the water of the ocean, or the difference, both as to magnitude and direction, between the attraction at the centre of the earth and at the various parts of the superficial ocean. The tangential tidal force at a point on the earth's surface having the angular distance $\theta$ from the moon is $\frac{r^3}{R^3}\cos\theta$ sin $2\phi$; and the radial force at that point is $\frac{r}{R}(\cos\theta+1)$; $r$ being the earth's radius, $M$ the moon's mass, $d$ the moon's distance from the earth, and $y$ the unit of gravitation. These forces are, then, inversely proportional to $R^3$. The differential tidal force is at its maximum directly under the moon, where it is all radial, and where it is only about 1/29th of the moon's whole attraction at the distance + +A diagram showing tidal forces acting on a canal around Earth. + +70 + +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +of the earth, or about 1/8,400,000th of $g$, or the earth's attraction at its surface. If the earth always kept the same side turned towards the moon, the lunar tidal forces would, of course, produce one tidal protruberance in the water on the side of the earth next the moon, and another on the opposite side. The protruberances would be stationary on the earth, and the discussion of their magnitude etc. would be one of hydrostatics only; they are therefore called static tides, or equilibrium tides. + +But as the earth rotates under the moon, the actual case in our equatorial canal would be very different. The two tidal protruberances and intervening depressions, in order to keep up with the moon, would have to sweep right round the canal in the mean period of time 50-60 minutes, at the rate of 1003-5 miles per hour. This is a velocity which after the passage of a tremendous torrent moving bodily along with that enormous velocity, but in the style of a smooth ground-oval in the sea, whose gentle wave-forms may be travelling onwards with a considerable speed, although the individual particles of the water are only moving backwards and forwards, for short distances, with quite different velocities. It is this difference between the active tides in our canal and reality of travel. We are therefore concerned with a dynamical question, and have to do, not with "statical", but with "dynamical", tides. The present subject is one on which it is very easy to go wrong; it contains several instances of what any person insufficiency acquainted with it would naturally regard, at first sight, as an apparent paradox. + +Let us begin by noting briefly the way in which the water moves in a travelling wave, or water-undulation. Anyone can observe this for himself when watching sufficient wind-waves on the sea; although such surface undulations differ importantly in certain respects from tidal ones, whose disturbances extend to all bottom of the ocean. See Fig. 18, which represents two waves moving towards the right. The upper dotted arrows + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +71 + +show the directions of the movement of the various parts of the water. The lower arrows show the directions of the gravitation forces due to the disturbance of level. On the crest of the wave the water is moving horizontally forwards with the greatest velocity; at the bottom of the trough the water is moving horizontally backwards with the greatest velocity. At the points of mean + +Fig. 13. + +level, halfway up the slopes of the wave-ridge, the water is moving neither forwards nor backwards, but on the front slope, vertically upwards; while proceeding to form the upper part of the ridge by addition in front; and on the hinder slope, vertically downwards; while withdrawing from behind binder of the wave-ridge, and thus forming a part of the wave, shown in a fore-and-aft vertical circle; in a tide-wave in a very elongated ellipse with minor axis vertical; this axis diminishing as we descend, until it vanishes at the bottom. The progress of the wave form is produced by continual addition of water in front, and subtraction of water behind. It is very easily seen that the velocity of the water, though entirely different from that of particles of the air, will be nearly exterior parallel, to the latter; and also that for a given velocity of the wave-form, its magnitude will increase or diminish in the same proportion as the velocity of the particles of water. + +Such a wave, having been started by some cause, would, on the cessation of that cause, continue to move onwards of itself, so long as it was not checked by any obstacle which was formed by the disturbance of level. There would be the unbalanced weight of the part of the wave projecting above mean level, and the unbalanced deficiency of weight in the part below mean level resulting in an upward pressure in that part, Fig. 13. It is evident that the said pressure and deficiency of pressure is + +72 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +proportional to the volume of water above, and deficiency thereof below mean level; that is to say (the oscillations being rela- +tively small), proportional to the greatest heights and depressions of the water. The forces are then always proportional to the distance from the position of rest; as in the case of a common pendulum oscillating with a relatively small linear amplitude; and the oscillations therefore are non-oscillatory, or periodical in their nature, and consequently the undulations will be always relatively small. Of course the forces will be, *exercia partibus*, proportional to $g$, the intensity of gravitation. If the deforma- +tion were so produced that the prominences and depressions, when left to themselves, would have no horizontal motion, the wave-forms (though not all the water) would simply oscilate up +and down, constituting stationary waves. But if started to move in any direction, they would continue to move them- +selves, in that direction, at their own rate, until their motion was destroyed by friction. + +The above-mentioned unbalanced weight and deficiency of weight in different parts of the wave, acts in a two-fold manner. +While the weight of the prominences tends to depress them, and by hydraulic pressure to force outwards the water of the im- +pinging parts below the moon's surface, while on the other hand the effect of gravitation on the more superficial parts of the water on the wave-dops is part of the whole motive force. The radial (or vertical) forces, whether downward or upward, and the tangen- +tical forces resulting from gravity conspire with each other in causing the movement of the water of a free wave; and there- +fore the general effect is the same in general character (which is all that now concerns us) as though the gravitation forces were entirely tangential. + +This is true of the lunar tidal forces also; the radial and the tangential conspire with each other in their constant effort to lower the water at 90° away from the moon, and to raise it under, side on the side from, the moon; their whole general effect is the same as if they were entirely tangential. This con- + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +Sideration is strengthened by the fact that the tidal effect of the lunar radial (or vertical) forces is quite insignificant as compared with that of the tangential ones. + +Therefore, considering what our present object is, we may, if convenient, treat both the gravitation forces and the lunar tidal forces as though they were wholly tangential; and it will be very convenient to do so presently. + +The two tide-waves with which we have to do constitute what we shall call an ellipse, at being nearly such; as represented by the ellipse in Fig. 14, which is Fig. 13 adapted to our + +Fig. 14. + +present purpose. They are supposed, in the diagram, to be moving, or revolving, relatively to the body of the earth (represented by the shaded circle), in the direction of the hands of a watch. The dotted arrows outside the ellipse represent the horizontal movements of the water itself; in accordance with what we have described above as the movements of the water in a wave. The arrows within the ellipse represent the positions and directions of the tangential forces acting on the water. The tangential forces are acting throughout one half of their reach, or extent, concurrently with, and through the other half against, the motions of the water which would be produced by them in a free wave; as well as all ordinary oscillations, for instance those of a common pendulum. + +Now it so happens that the general scheme of the lunar diffe- + +A diagram showing an ellipse representing two tide-waves moving relative to the body of the Earth. Arrows indicate horizontal movements of water and tangential forces acting on it. + +74 + +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +rential forces all round the earth, as regards their positions and directions relatively to each other, is similar to that of the above-mentioned gravitation forces; so similar that if the moon be supposed to be opposite a side of the tidal ellipse, the members of the two sets of forces will, with a trifling exception mentioned below, respectively act upon each other in the same direction. The diagram of the tidally-elevated water, shown by the inner arrows in Fig. 14, produces, as we have seen, the motions of the water shown by the outer dotted arrows in that diagram. It is evident, then, that the lunar tangential forces, whose scheme is similar, if they could act by themselves, without calling into being the gravitation forces, would produce, under the condition of the rotation of the earth beneath the moon, exactly those effects which would be produced by these three actions would be represented by the said outer arrows in that diagram, and whose relations would be very nearly those of the different parts of a great ocean wave whose length was equal to a semi-circumference of the earth. + +Thus the actual tidal waves move under the influence of a scheme of lunar forces, acting along with a generally similar scheme of gravitation forces, such that both themselves have occasioned. (In the present chapter we are quite unconcerned with the trifling differences of detail which exist between the lunar and the gravitation forces. The only one worthy of mention is that whilst the very slightly operative lunar radial (or vertical) force vanishes at 54° 44' from the moon, the gravitation radial disturbance varies from the mean level of 0.6 mm., whilst this frictional would be 40 cm. from the moon, very nearly, and, with friction, differently situated, as will be seen from pages 81 and 82 below.) + +One considerable difficulty in understanding the production of the dynamical tides arises from the coexistence and cooperation or antagonism, as it may be, of these two systems of forces. + +Let us enter now that if $v$ be the velocity with which a free, frictionless undulation of the water, reaching to the bottom + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +and of very great length relatively to the depth of the water, +would travel, of its own accord, $v = \sqrt{g}d$, $d$ being the depth of the water and $g$ gravity. In order that such undulation should so travel with the mean velocity necessary for its keeping up with the moon, at the equator, viz.: 1005 miles per hour, the depth of the water ($= v^2/g$) would be 1276 miles. As there are two complete tides in every lunar day of about 24 hours 50 minutes the mean period of a single tide would be 12 hours 30 minutes. Secondly, if the depth of the water were less than that just mentioned, a free tidal wave could not keep up, of itself, with the moon; and its period of oscillation would be greater than 12 hours 25 minutes; if it keeps up with the moon, as it would have to do, it must be as a “forced wave,” forced by lunar tidal action. But if the depths were greater than the depth just mentioned, a free tidal wave could not keep up, of itself, with the moon, as it would have to do, it must be again as a forced wave, but one whose velocity is restrained by the lunar tidal action. The depth now in question we shall call the critical depth. (That is for the equatorial canal.) If the canal ran along the parallel of latitude $\lambda$, the velocity necessary for keeping up with the moon would be 1005.5005 miles per hour; and the critical depth would be 1276.005 miles.) + +What then will be the position of the lunar dynamical high tide, relatively to the moon? + +This is really a manifold question, which requires four different answers: viscosityless water is supposed to be with, or without, viscosity or friction; and as the depth of the water (always uniform) is supposed to be less, or greater, than the critical magnitude just mentioned. We shall consider afterwards the case when it is of that magnitude. + +Firstly, let us suppose that there is no friction, or viscosity, in the undulating water. + +A 1. Let the depth of the frictionless water be less than 1276 + +76 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +miles, the critical depth, so that a free tidal wave would oscil- +late more slowly, that is, with a greater period, than the forced +tidal wave. In this case low water of the dynamical tide will be +under the moon; that is, high water (which for the statistical +tide would be under the moon) will be 0° behind, or east of, +the moon. + +A 2. But let the depth of the frictionless water be greater +than 12-76 miles, the critical depth, so that the free tidal wave +would oscillate more rapidly, that is with a shorter period, than +the forced lunar tidal wave, then high water of the dynamical +tide will be under the moon; that is, it will occupy the same +position, relatively to the moon, as high water of the statical +tide. + +Both these cases are comprehended in Airy's general mathe- +matical expression for the height of the water of the frictionless +dynamical tide in an equatorial canal, at a given angular distance +from the moon. (See Note A.) + +Airy proposes the following interesting illustration of this:— +If there were two equatorial canals, such as the above, side by +side, to all appearance similar, one, however, being less and the other more deep than the other, then, with frictionless +dynamical tides, high water in one canal and low water in the +other would run abreast. (See Note B.) + +We can, for ourselves, put the explanation of this into the +following simple form, which will be found to be quite sufficient; +although it does not go into any details of the movements of the +water. + +[X.R. We shall sometimes, for brevity, speak of water which is +less than the critical depth as " shallow" water, and of that +which is of greater, as " deep" water.] + +Let us begin with considering a simple example which illus- +trates the general principles involved. + +In a pond or a large lake or sea, if it would have its own proper +period of vibration under the influence of gravity. Now suppose + +**RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY.** + +It is to be noted upon by a system of small reciprocating impulses which have a different period, and whose magnitude is inde- +pendent of the amplitude of the vibrations and constant, the forces of the impulses varying between zero and maximum accor- +ding to their own law, and symmetrically on each side of the point of rest of the pendulum. The amplitude of the vibrations will increase by accumulation, and the tangential gravitation force called in being by the impulses will diminish from the positive to negative proportional thereto, until they become great enough to be able, by the baffling effect due to their efforts to establish their own vibration period, to prevent any further increase in the amplitude of the vibrations under the small external reciprocating impulses, which, as we have said, remain of constant magnitude. It is evident that the smaller the difference between the periods of the two systems, and that of the impulses, the less will be the said baffling effect, and the greater the final amplitude of vibration. When the amplitude has arrived at the maximum (equal on both sides of the point of rest) for the given pendulum and for the given reciprocating impulses, the final, settled state of things is reached; the period of vibrations being that of the impulse. The two systems of forces are then in equilibrium with respect to position of rest of the pendulum, and therefore so with each other. + +So must it be with the scheme of gravitation forces created by the tidal deformation of the surface of the water of the equatorial canal and the scheme of the lunar tidal forces. They must get into such a final relative position that their respective axes of symmetry will coincide; and this is certainly involved in the coincidence of their axes when all impelling those forces are disturbed lunar forces; leaving the question still to be settled in which of the two possible ways the coincidence will occur in the particular case; whether as in Fig. 15, or in Fig. 16, the moon being away to the right. Either the longest or the shortest axis of the tidal circle must point directly to the moon. As before, the shaded circle is the body of the earth, and the + +A diagram showing two pendulums with different periods and amplitudes. + +78 + +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +ellipse the surface of the water, the ellipticity being enormously exaggerated. The apparent motion of the moon, or that relative to the surface, is always watch-wise. + +Take now the case of a "shallow" water tide in the equatorial canal, which would spontaneously oscillate and travel more slowly than the moon would do. As we have said, the motive force, causing the spontaneous free oscillations of the + +Fig. 15. + +water, is the weight of the high-tide prominences and the deficiency of weight of the intervening low-tide depressions. Now it is evident that the tidal wave has to travel at the moon's rate; however this be brought about. In order that the "shallow"-water tide may oscillate and travel quickly, let us suppose that it has been placed in a situation remote from the moon so that its own just-mentioned motive force shall have the moon's tidal forces helping them; and it is evident that this will be so when the middle of a side of the tidal ellipse is, at least, nearly opposite to the moon. That is to say, low water must be, at least, nearly under the moon; and from what we have seen above, if it be nearly so, it must be directly so, as in Fig. 15; and this of course applies equally to all depths of water less than the critical depth. + +Take now the case of a "deep"-water tide, which would, if + +**RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY.** + +free, oscillate and travel more quickly than the moon would have it to do. It must travel at the moon's rate; however this be brought about. In order that it may move slowly enough to keep pace with the moon, it must get into such a position relatively to the moon that its own motive forces shall have the same effect on it as those of the moon. This will be so when the end of the tidal ellipse is, at least, nearly opposite to the moon. In other words, high water must be, at least, nearly under the moon, and therefore directly so, as in Fig. 16, and this manifestly applies equally to all depths of water greater than the critical depth. (See Note C.) + +Fig. 16. + +Thus the summit of a "deep" water dynamical tide would occupy the same position, relatively to the moon, as that of a statical tide. But the magnitudes of the tides would generally differ. If the water were not too much deeper than the critical depth, the dynamical tide would be the greater; but if the water were deep enough, the statical tide would be greater and in course for a certain intermediate depth they would be equal. + +We may here note the following:---If two canals of uniform width and depth ran side by side along two parallels of latitude not too close together, each returning into itself, and if they were both of the critical depth corresponding to the mean latitude $\lambda$, which depth would be, as we have seen, $1278$ cos$\lambda$ + +79 + +80 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +miles, then high water of one canal and low water of the other would run abreast; since the more northerly canal would be deeper, and the more southerly shallower, than its own critical depth. + +Perhaps it might be thought that if the water, as under the present supposition, were absolutely frictionless, there would be nothing to prevent the perpetuation of a stationary tide sweeping round the earth in this manner, so long as it kept up the angular rate of the moon, with its summits always under, and on the off side from, the moon. It is quite true that such a tide of the proper magnitude formed and set going by some other agency so to travel with perfect accuracy, would be kept up by the moon, and would preserve its position relatively to the moon. But the moon itself could not so start such a tide; because the centrifugal force of its motion would tend to throw out undisturbed water on the rotating earth, which would produce therein, at once, a system of varied movements agreeing very nearly with that of a free wave-motion (see p. 74); thus creating imme- diately a dynamical tide. + +Now let us recognize the friction, or viscosity, of the undula- ting water. + +B 1. When the depth of the water is less than the critical depth (so that a free, frictionless, tidal wave would move more slowly than the moon would have it do), the effect of the addition of friction, paradoxal as it might seem at first sight, is no less than to make the tidal wave fall back to the moon, that it would occupy without friction ; that is to say, high tide would be somewhat less than 90° behind, or east of, the moon; and it would occur sooner in time than it would for frictionless water. + +B 2. On the contrary, if the depth of the water were greater than the critical depth (so that a free, frictionless, tidal wave would move more quickly than the moon would have it do), the effect of added friction would be to make the point of high + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +81 + +water to be behind the place, relatively to the moon, that it would occupy without friction; that is, it would be a little dis- +tance behind the moon, instead of being directly under her, as it would be without friction, and it would come later in time than for frictionless water. + +The analytical proof of these two statements will be found in pp. 331* and 532* of Airy's Art. referred to in Note A, +though he does not give the place of stated facts. It is intended to explain us to do this for ourselves. +We may observe that all four statements A1, A2, B1 and B2 will be found in Prof. George H. Darwin's Art. "Tides" in the last edition of the Encyclopaedia Britannica. + +The geometrical proof of B1 and B2 is quite simple. For it we must now turn to the consideration of the movements of the different parts of the water of the tidal wave; for it is on this that friction depends. + +First we take case B1, availing ourselves of the mode of proof given by Rev. T. K. Abbott, Fellow of Trinity College, Dublin. + +Fig. 17. + +A diagram showing a dotted line labeled "Is Mass" and an arrow labeled "A" pointing upwards. + +Suppose that we are standing on the ground beside the canal at a, Fig. 17 ; the body of the earth rotating under the moon counter-watch-wise; as we are carried on towards the point under the moon, the velocity of the tidal current indicated by the dotted arrow is increasing under the continued action of the lunar tangential force indicated by the broken-line arrow; and therefore the frictional resistance due to the current is increasing o + +82 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +in the opposite direction ; in addition to this, the tangential lunar disturbing force, which has been, and is, giving the water its increasing velocity, is itself diminishing. The friction-resistance will therefore become equal to the oppositely directed lunar tangential force, somewhat before this force becomes zero ; that is, at a point short of that under the moon. +At that point, then, +the whole tangential force passes through zero, and changes its direction, beginning with the part behind the moon, and then behind it ; thus ensuing it to cease falling sooner than it would do without friction, and at a point ahead of that under the moon. +And, for a similar reason, high water will occur at a point short of, i.e. ahead of, 90° behind the moon. Thus, as in Fig. 17, the axis-minor of the tidal ellipse will not point to the moon. This acceleration of the phase of the tide is evidently at the expense of some retardation of the water on its way back again. For before it has reached what would be its lowest point without friction. + +In case B2, as we can easily see for ourselves, the contrary takes place ; because the directions of the tidal currents, both under the moon and 90° away, are the opposite of what they are in case A1. Suppose that we are standing on the ground beside the canal at b, Fig. 18, which has not yet reached the point + +Fig. 18. + +under the moon. As we are carried by the rotation of the earth near to that point, the lunar tangential force is slowing the current ; and the friction-resistance, now near its maximum, compares with it in so doing. When we have reached the point + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +under the moon the lunar tangential force has vanished; but the current, of course, continues, and the friction continues slowing it, and though the lunar tangential force begins, at that point, to act in the opposite direction, and against friction, it will not become equal to it (and then greater than it) until we have been carried by the rotation of the earth more or less behind the moon, as in Fig. 18. Therefore the greatest slowing at a point behind that under the moon; and there high water will occur; and on the other hand, low water will be similarly retarded; and, as in Fig. 18, the axis-mover of the tidal ellipse will not point to the moon. + +Or thus—Friction, in the case of a "shallow" water tide, prevents the full formation of the hinder parts of each tidal prominence, and of each tidal depression; thus making the high-tide and low-tide points of each tidal prominence lower than they would occupy without friction; but this, as is evident, is at the sacrifice of some of their height and depression, respectively. + +On the other hand, in a "deep" water tide, the friction prevents the full formation of the front part of the tidal prominence and of the depression, with, of course, a contrary result; and again at the sacrifice of some of the height and depression of the tidal wave itself. + +It is evident that the forward, or backward, shift of the position of low, and of high, water will be greater, ceteria paribus, for a greater coefficient of friction; though only up to a certain limit be mentioned later on, p. 58. It will also be greater, ceteria paribus, for a nearer approach to equality between the periods of free, and of forced oscillation, or for a nearer approach of the depth of the water to the critical depth. + +But now let us ask, if the frictionless water were just of the critical depth, what would be the position of high water relatively to the moon? The principle appealed to above will help us to answer this question as well as it can be answered. The moon would not then be forcing the water to oscillate, and the total + +9 + +84 + +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +wave to travel at a rate different from its spontaneous rate under gravitation; and therefore the lunar tidal forces would have to be, on the whole, neither helping nor opposing gravity, as regards affecting the oscillation-rate of the water. It might, then, seem that if there were a tidal ellipse, the four points of half tide, where the water is at the mean level, should be at the points under and opposite to the moon and 90° before her and behind her; but this is not so. The tidal ellipse generally would be under the moon, and about 45° behind the crest of the moon, but the case would be a peculiar one. If the depth of the water were ever so little less than the critical depth, high water would be 45° behind this said point; and if the depth were ever so little greater than the critical depth high water would be 45° before that point. Thus, even though it were possible to make such an ellipse, yet because high tides should remain at 45° behind the moon, it would not be practically possible; because the condition would be one of instability. But, moreover, even if a tidal wave in water of the critical depth could be, by some means, formed with its crest 45° behind the moon and started so as to keep up with her, the lunar tangential force (veryly more important than the radial) would be, as is evident, constantly opposing gravity in that part of the wave which is moving towards it finally in his direction. The result would be such confusion as would destroy the long wave before long. We have just seen that it would be impossible for the crest of the wave to remain at any other point within the first quadrant behind the moon; it must be either 90° away from the moon or under the moon, with an equal right to both positions. Since no perturbation can affect this position without being at once equal to that of the free oscillation of the water, it is evident that those oscillations, if they existed, would become infinite, but for certain conditions of the case which would prevent this. + +The above conclusion, drawn from simple geometrical considerations, is in accordance with that which Airy derives from his equation given in Note A, which see. He observes that if and + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +m become equal, that is if $d$ and $e$, in our simplified form of the equation, be equal, that is if the water be at the critical depth, the tide would be theoretically infinite, and the equation fails. +His interpretation of this failure is that the motion of the water would not be oscillatory in the manner of a wave; but that it would be that of a torrent of unequal depth passing round the earth so as to follow the apparent motion of the moon. +The reason why this is so, is that since frictional heat keeps the height of the tide finite, even though the water were of the critical depth. Since friction always increases with the velocity of the oscillating water, which velocity would obviously increase with the magnitude of the tide always going at the moon's rate, friction would increase with the latter (see same Note), and therefore the magnitude of the tide with friction could not exceed that without friction. Hence, while the velocity of the increasing friction in keeping down the magnitude of the tide became equal to that of the lunar forces in accumulating, or piling it up. + +It is generally considered that in the actual, relatively very small, tides of ocean (away from shores), because of the smallness of the velocity of the particles of water, the friction is very nearly proportional to their square of velocity. But in the present supposed case, in which the tides would be very much larger, and in which the velocity of the water would be correspondingly great, the friction would be probably nearly proportional to the square of the velocity; and as the forces of free oscillation would be very nearly proportional to the distance of the summit of the waves from shore. In such cases, it would seem probable that friction, in this case, would be very nearly isochronous, for different amplitudes, and moreover their period very slightly altered (in accordance with a well-known dynamical principle illustrated by a pendulum with small amplitude of oscillation, whose period is sensibly unchanged by the resistance of the air, if this varied as the square of the velocity). Thus, while the friction would keep down the magnitude of the tide within reasonable limits, it + +S6 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +would alter very little the period of the free undulation of the water; and consequently the critical depth of the water with such friction would be nearly the same as that for water without friction. + +Suppose the depth of the "shallow" water to increase gradually, the magnitude of the tide will increase faster than the depth; and therefore the velocity of the water will increase, and all will be raised (see Note A); but this will cause a shift of the point of high water, which will be higher. But as the frictions prevents the height of the tide, and therefore the velocity of the particles of water, from becoming indefinitely great, it indirectly prevents its own self from becoming so; and therefore the summit of the tide could never get within a certain distance of the point of 45°, even though the water attained to the critical depth ; said distance being equal to that between the points of 0° and 45°, which is a measure of friction. Similarly, if a "deep"-water canal shallowed gradually to the critical depth, the summit of the tide could never get within the same distance of the point of 45°; and the limits between which it would be impossible for the high tide to remain would be much closer than before. + +We have seen that, in the "shallow"-water tide, acceleration of the various phases of the tide is, ceteris paribus, greater as the coefficient of friction is greater. But it will be easily seen, on consideration, that no amount of friction in a "shallow"-water tide would be able to make the angular displacement of its highest point less than 45°. For if at any time during which the water is of less than the critical depth, the moon must be forcing the tidal wave to travel faster than it would do if itself; she must be, on the whole, working with the gravitation forces to accelerate the oscillations of the water; and, as is evident, she will not be doing this unless the end of the tidal ellipse is more than 45° behind her. To this we may add that the confusion mentioned in Note B is important. If the crest of the tide were sufficiently near the point of 45°, and would help in preventing its reaching that point. + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. S7 + +A corresponding statement, *mutatis mutandis*, is of course, to be made respecting a "deep" -water tide. No amount of friction would be able to make its high water fall back to 45° behind the moon. (See Note D.) + +It is important, for certain reasons which need not now be mentioned, to note particularly the conclusion from the above--what indeed has been already stated by Prof. G. H. Darwin, who says, "The dynamical tides are always under the moon, and whether there be, or be not, friction, the crest of that dynamical tide whose position, if it were a statical tide, would be under the moon, can never be outside the first quadrant behind the moon; and that, if there be friction, it must always be within that quadrant. + +It is, perhaps, more important, for reasons which need not be mentioned at this note particularly that, as we have seen, whatever the depth of the water, and whether there be or be not friction, the crest of the dynamical tide cannot be at the point of 45° behind the moon. + +All the above, of course, applies equally to the solar dynamical tide in an equatorial canal; except that for this tide, whose period is 12 hours, and whose rate of progress would be about 10574 miles per hour, the critical depth ($m = \frac{c}{v}$) would be greater, viz., about 1367 miles; and also that, as *tertia pars*, the friction of the smaller solar tide would be evidently less than that of the lunar in a higher ratio than that of the respective tidal forces; and therefore (which is a very important reason), the shift of the points of high and of low water, on account of friction, would be less than that for the lunar tide. + +Airy points out the interesting conclusion that if the depth of the equatorial canal were between the lunar and the solar critical depths, that is between 12'76 and 13'67 miles, and there were no friction, since high water of the lunar tide would be under the moon's quadrant and high water of the solar under its quadrature, both tides would concur with the quadratures, and neap tides with the syzygies, of moon and sun ; or the reverse of what now obtains. + +88 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +To this we may add, for ourselves, the following respecting the position of high water of spring and of neap tide, when the actual depth of the Frictionless water is between the two critical depths. High water of spring tides would always be under the moon, and low water would be over the sun; but the position of high water of neap tide would depend on circumstances. As long as the lunar tide was greater than the solar, high water of neap tide would be under the moon and sun. But if the actual depth were sufficiently nearer to the solar than to the lunar critical depth to make the solar tide greater than the lunar, then high water of neap tide would be 0°0' behind sun and moon. + +Note A, from p. 70, *Airy's equation* for K, the distance from the mean level of the surface of the Frictionless dynamical tide in a uniformly deep and wide equatorial canal entering into itself, may be found in his Art. on *Tides and Waves in Encycl. Metrop., vol. v., p. 322*. It is, after setting aside a certain term which is relatively quite insignificant, +$$K = \frac{H}{\sin^2 \theta} \cdot \frac{n^2}{1 - \cos^2 \theta (d - m \cos \theta)}.$$ (A) + +If being the moon's tangential tidal force at its maximum (or $g/11,800,000$, which it attains at 45° away from the moon; as being $\frac{1}{2}$ divided by the length of the wave, which length, at the equator, is the circumference of the earth), making $m = \frac{2}{\sqrt{3}}$ at the equator (R being the earth's equatorial radius); $\iota$ is to inversely as the period of the forced wave (or $12^{\circ} 20^{\prime} 5''$) to that of the free wave, for the actual depth of the water (which is defined by $d$), and $d - m \cos \theta$ is $\frac{1}{2} R$. $\theta$ being angular distance of the point in question from the moon. Therefore the above equation can be written thus, in a form more convenient for our present purpose, giving the value of $K$ in feet : +$$K = -0.00 \cdot \frac{d}{c - d} \cos 26^{\circ}.$$ (B) + +RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. + +This expression, like $(A)$, from which it follows, is only approximate; and therefore for certain purposes it would not be right to give it too great a range of application. But the following can be derived from it. + +Selecting the point under the moon where co $2\theta$ is 1 and a maximum value of the numerator, whether positive or negative, if the depth of the water be less than the critical depth, the denominator is positive, and $K$ is negative; i.e. low water is under the moon; but if the depth be greater than the critical depth, $K$ will be positive, and high water will be under the moon. We see also that if the depth be small relatively to the critical depth (but only on that condition), the height of the tide varies nearly as the depth; that is, in a slightly higher ratio than in a higher one; but when the depth exceeds the critical depth, $K$ is theoretically infinite, and the expression fails. + +We may note also the following—It is easily seen that if the depth of the water were to increase gradually, and if the magnitude of the tidal wave increased in the same proportion, the velocity of the particles of water would be constant, and the frictionless condition would hold. But if the depth of the water were increased, the magnitude of the tide would increase in a higher ratio, and therefore friction would increase. + +We see also from equation $(B)$ that if the actual and the critical depths be not very different, the height of the tide will vary nearly as the inverse of the difference. Hence, if the actual depth were between the lunar and the (greater) solar critical depths, or if it were greater than those depths, the lunar tide would show about $\frac{1}{2}$ of the lunar, would be greater of the two, if the actual depth were sufficiently nearer to the solar than to the lunar critical depth. This, however, may be called self-evident after some of the considerations adduced in the text. + +Note B, from p. 70—in Airy's well-known geometrical proof of the position, relatively to the moon, of the frictionless + +A diagram showing a celestial body with a line representing its critical depth at various points. + +90 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE + +dynamical high tide (M. Note., R. A. S., vol. xxxvi, p. 229), the writer had his attention fixed exclusively on the case in which the moon must be forcing the reluctant water to oscillate fast enough for the tidal wave to keep up with herself, and thus the demonstration applies only to "shallower" water tides. Conse- +quently in that place the reader has only one side of the question at +set forth to him, and if he is unanimous about the author's +analytical treatment of this subject, he will probably find part of +the very paper now cited, or in the Art. referred to in our last Note, +he may be (and apparently sometimes is) left under the im- +pression that low water of the frictionless dynamical tide is +necessarily always under the moon. But this geometrical proof +can be easily applied, mutatis mutandis, to "deep" water tides, +if we remember that in them the moon must be forcing the +water to oscillate faster than it does in the tidal waves, when +produced, may keep back with herself. + +Men. It is easily seen that the equation for K, the height +of the frictionless dynamical tide, given by Airy in p. 229 of the +paper referred to in this Norr., and in Enc. Metr. vol. v. p. 323 *, is +$$K = \frac{1}{2}H_0 - \frac{1}{2}\pi\mu g\cos(\theta - m\varphi),$$ +the same as that (marked A) which we have copied in our last Norr., although they look so different. + +Norr. C, from p. 78.—It might, perhaps, seem at first sight +that if the lunar relations of direction between the various +movements of the water and the scheme of lunar forces were +consistent with the moon's keeping up the tide, the reversal of +these relations should be inconsistent therewith. But let us +remember that in the former case the lunar forces were acting +for only half their time concurrently with, and for the other +half against, the movements of the water, just as with all +ordinary oscillations or vibrations. We are no worse off now +as regards this than we were before; only the difference is that + +**RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY.** + +the concurrences and oppositions have exchanged their situations relatively to the moon. + +Though it involves a little repetition, we may take this oppor- +tunity of putting together the answer to the following point, +which some might possibly feel, at first sight, to be a difficulty. +The spontaneous movement (if permitted) of the tide, when +created, would be due to $g$; how then can the lunar tidal forces +act on all the water throughout its whole depth, $h$, or even +to a very small proportion of it? Because the lunar tidal forces act on and move all the water throughout its whole depth, +and are sensibly independent of the existence of the tidal de- +formations; while the gravitation forces, though acting on all +the water, are self-balanced as regards their pull on that below +the level of low water; the forces which would produce the +deformation being so small compared with those of $g$, and the +gravitation forces acting on the relatively very small superficial +tidal protruberances, and are proportional thereto. Suppose that +a wave like the tidal wave, and even of great magnitude, +whether in "shallow" or in "deep" water, were created and started to move from E. to W., by some other agency, and then left to the moon alone; the confusion due to the continued +changing between these two agencies, and also between them +others, at first very much larger, would reduce the magnitude +of the tidal deformation until the gravitation forces, dependent +on, and proportional to, that deformation, became diminished enough to be under the control of the independent and constant lunar forces. + +*Norm D*, from p. 87.—This follows also from Airy's equation +for the height $K$, above mean level, of the surface of the tidal wave with friction, at the angular eastward distance $\theta$ from +the moon. See *Enc. Metr. vol. v., pp. 351* *et seq.* If the +wave is $w$, then $K = \frac{w}{\cos \theta}$; but if the equation at the very bottom of p. 351 *et seq.* may be written thus: + +$$K = \frac{w}{(1 + tan \theta) \cos (2\theta + D)}$$ + +92 +ON THE POSITION OF THE DYNAMICAL HIGH TIDE. + +C being a constant, and D an angle whose tangent is proportional to the coefficient of friction, which would be the same at every part of the tidal wave for any given friction, but would vary in the same direction as the friction (were this to alter) ; the upper limit of the magnitude of this angle being 90°; that is, if the water be more than the critical depth, then (D = 29° + D = 180°, or $\theta$ = 90° - J.D.). If, now, the friction be so exceedingly great that D is nearly up to its maximum 90°, then $\theta$ (which without friction would be 180°) is slightly more than 45°. That is to say, the forward shift of the point of high water, due to exceedingly great friction, must be less than 45°. + +If the water were of more than the critical depth, the equation would be + +$$K = \frac{C}{(1 + \tan^2\theta)(29 - D)}$$ + +from which it follows similarly that the backward shift of the point of high water, from exceedingly great friction, must be less than 45°. + +These equations show also (what indeed is self-evident) that if the friction were very great, so that D was not far from 90°, making tan D very large, the magnitude of the tide would be exceedingly small. + +N.B.—We omitted to explain that in Figs. 15 and 16 the arrows outside the ellipses represent the lunar tidal forces, the moon being to the right; those within the ellipses represent the gravitation forces of the disturbed water. + +[ 93 ] + +CHAPTER VI. + +THE "HORIZONTAL" PENDULUM. + +ALTHOUGH the moon's differential tidal force is quite easily calculable, and its magnitude perfectly well known, various attempts have been made to detect it by direct observation. The most important, but not the earliest, of those was carried out by Sir George Airy, F.R.S., in 1850, under the auspices of the British Association, consisting of Professor George H. Darwin and others. The description of the apparatus used and of the experiments made therewith is given in the Brit. Assoc. Report for 1881. The attainment of the same object had been before sought by means of what is called the "Horizontal" Pendulum *. This is a simple instrument intended for the measurement of very small changes of level in a horizontal plane, for the detection of exceedingly small changes of level in the platform on which it stands. It is capable of very much greater sensibility, as regards the latter, than the most delicate spirit-level ; moreover, its sensibility can be quite easily regulated in accordance with requirements. + +Apparently the first to set up such an instrument was Hengelbrg, a pupil of Grünthüsen at Munich, who, not later than 1832, did so in the manner shown in fig. 10. (See paper by Prof. Safirik in Phil. Mag., vol. xlvii., 1873, p. 412.) de is a rigid rod carrying at its end a ball of metal. The wire cu is + +* This name is useful as a designation only, not as a description. The Pendulum's rod need not be horizontal, and its plane of oscillation must not be so, if it is to be a gravitation pendulum. + +A diagram showing a horizontal pendulum apparatus. + +94 +ON THE "HORIZONTAL" PENDULUM. + +attached at one end to the rod, and at the other end to a point of support $a$; the wire $db$ is attached at one end to the rod, and at the other end to a point of resistance $b$. The imaginary line joining $a$ and $b$ is nearly vertical, but leaning slightly + +Fig. 19. + +A diagram showing a pendulum with a horizontal axis. Point \( e \) is shown as a small circle near the top left of the diagram. Points \( a \), \( b \), and \( d \) are labeled on the diagram. Line segments connect these points, forming a triangle. The line segment connecting \( a \) and \( b \) is nearly vertical, but leans slightly towards the pendulum. + +towards the pendulum. Of course the smallness of the distance \( ed \) does not contribute in the least to the sensitivity of the instrument; it would do so only if the wires \( ae \) and \( bd \) were both always kept parallel to each other, which they are not on account of the axis \( ab \) to verticality. The horizontality of the pendulum-rod is of no importance, except for convenience. It might slope upwards or downwards from \( d \) at an angle of 45°, if desirable, without affecting the working of the Pendulum. + +It has been stated that Gauss set up such an instrument. It is very likely that this is correct, but in its absence we do not see any experimental demonstration. It is possible that the statement may be founded on a confusion between the bifilar pendulum now in question and Gauss's bifilar magnetometer, which, however, acts in a quite different manner. + +About 1531, Mr. Alexander Gerard, of Gordon's Hospital (now College), Aberdeen, suspended such a pendulum in the manner represented in fig. 20. His account of it will be found in Edich, New Phil. Journ. for April 1631. de is a rigid rod + +ON THE "HORIZONTAL" PENDULUM. +95 + +pointed at the end $d$, the point being of steel resting against an agate cup in ad, the side of a stiff standard. The thread ac is attached at one end to the rod at its centre of gravity and at the other end to the standard; ad should, of course, lean very slightly towards the pendulum. + +Fig. 20. + +A diagram showing a right-angled triangle with sides labeled a, b, c, d. + +In 1862, M. Perrot did the same, and exhibited his instrument to the French Academy. His mode of suspension, shown in fig. 21, was the same as Hengelier's, with, however, this difference, that the supporting threads ac and bd were acting very much less nearly against each other, the advantage of which is obvious. We shall return to this subject. Of course cd is less than ac. Perrot's description is in Comptes Rendus, vol. liv., March 21, 1862, p. 728. + +96 +ON THE "HORIZONTAL" PENDULEM. + +In the early part of the year 1869, Rev. M. H. Close, of Dublin, suspended such an instrument in the manner shown in Fig. 22. $de$ is the pendulum-rod, and $ac$ and $bd$ the supporting threads attached at $a$ and $b$ to a stiff standard leaving very slightly towards the pendulum. (See Practical Physics, by Prof. W. F. Barrett and Mr. W. Brown, London, 1882, p. 213.) Of course the threads, as also those of Gerard and Perrot, had better not be twisted threads, which are liable to be affected by + +Fig. 22. + +the hygrometric state of the air. In this Fig. $at$ is an accurately straight and smooth edge projecting from the supporting standard towards the spectator, across which edge the separated silk fibres of the threads (which cannot be shown individually in the diagram, on account of the smallness of its scale) are bent at $a$ and $b$, so as to be practically attached thereto. The fibres are made to cross the edge separately, and close together, so that each one is supported by two edges only, in such a way that each may hang freely the sum of what we may call (in analogy with "tension") the "flections" of the several fibres, without the tensions of the outer ones which would exist if they crossed the edge in a single cord. In this case there is no tension of the supporting threads from the movement of the pendulum. + +In the same year 1869, and, to judge from his own words, in the middle part of that year, he invented a new unknown "Horizontal" Pendulum; his mode of suspension being the + +ON THE "HORIZONTAL" PENDULUM. +97 + +same as Hengelser's with, however, the difference that the heavy bob of the pendulum was placed quite near the axis of oscillation. See Fig. 23. His description of it will be found in a paper "On a new method for the Measurements of Attractive and Repulsive Forces," in the Proceedings of the Royal Saxon Soc. of Sciences, Nov. 27, 1850. He describes it also in Phil. Mag. vol. xlii, 1872, p. 401, giving a drawing in plate 3 of that volume*. The wires of Hengelser and the threads of Perron are thin wire-springs, each about 11 inches long and attached above to a horizontal bar, upright column, or standard, nearly two inches in diameter, supported on three feet and furnished with delicate levelling screws. The + +Fig. 23. +whole height of the stand being about 32 inches. The cylindrical bob, made of lead, and of about six pounds weight, carried in front a mirror by which readings were made on a reflected scale, according to a general modern practice. This pendulum is superior to Hengelser's, in that, for a given weight of the whole pendulum, the strength of the support is greater, and therefore there is less this unavoidable stiffness. + +* It is somewhat unfortunate that "Horizontal Pendulum" does not occur in the Index of this volume. + +II + +98 +ON THE "HORIZONTAL" PENDULUM. + +much less; and also in that the weight of the bob has, for a given angular departure of the pendulum from the position of rest, so much less sideward moment against the supporting structure, which is calculated to cause lateral yielding thereto. +Lord Kelvin's device for attaining the same object as above is described in the Brit. Assoc. Report for 1881, p. 203; it is in reality a Horizontal Pendulum. + +Dr. von Rebeu-Pasewitzki's Horizontal Pendulum may be described as similar to Zöllner's; except that instead of working by the torsion of elastic bands it turns on pivots at a and b, Fig. 23, consisting of stout points in agate cups. It is described and figured in the Report just quoted for 1883, p. 305. This instrument has the great advantage of being free from the influence of gravity, and of being capable of being supported &c., but it has, like others, its own special disadvantages which need to be guarded against. + +The Gray-Milne seismograph, suspended in 1891 by Prof. Gray, of Terro Hante, Indiana, U.S.A., and Prof. Milne, of Tokyo, Japan, is a Horizontal Pendulum on the general plan of that represented in Fig. 20; see description and diagram of it in the Report for 1893, p. 107–108. + +Mr. Horace Darwin's Biiflar Pendulum, described and figured in the Report for 1893, p. 201, is a Horizontal Pendulum on the general plan of that in Fig. 22, above. + +Of course such instruments are read, whenever practicable, by means of a scale reflected in a mirror attached to the pendulum; but this is not always convenient when at points, say r.e. and t, Fig. 24, forming a right angle at t' there being levelling screws with graduated heads at r.e. and s. If the pendulum be in the position pp', would be the regulating-screw for determining the lean of the axis of vibration towards the pendulum, and so adjusting the sensibility of the instrument; and s would be the setting-screw for setting the pendulum to zero, when necessary, or for testing the sensibility. + +ON THE "HORIZONTAL" PENDULUM. + +It is very easy to see that, neglecting proportionally the force of torsion of the supporting threads, wires, &c., if the inclination $\theta$, of the thread to the vertical be very small and the angle tilt to be measured be also very small, then the measure of the instrument must have special superiority without this, the delivery of the instrument is proportional to $g/\sin\theta$, or inversely proportional to $\sin\theta$, or to $\theta$ itself. For a given very small angular + +Fig. 24. + +A diagram showing a pendulum with a string attached to a pivot point at the top, labeled 'P', and another string attached to a mass at the bottom, labeled 'm'. The strings are labeled 's' and 'r' respectively. + +change $e$ of the surface of the ground, transverse to the vertical plane of rest of the pendulum, the angular movement of the pendulum would be magnified to $e/\sin\theta$. + +But the force of torsion of the supporting threads, or wires, &c., diminishes the magnification of the tilt to be measured and the sensibility of the instrument. For small departures of the pendulum from its position of rest, this force of torsion and the temporal component $e$ of $e$ (or $m$ - pendulum's mass) vary somewhat according to the same law, viz. $e$ varies as the distance from the point of rest. Therefore the pendulum is oscillating, not merely under $m\sin\theta$ acting at the centre of mass, but under $m\sin\theta + r \cdot e$ being the magnitude of the force of torsion of the supporting threads, or bands, as acting at the centre of mass of the pendulum, or the moment of torsion + +w2 + +100 + +ON THE "HORIZONTAL" PENDULUM. + +divided by the distance of the centre of mass from the axis of oscillation. Consequently if the stand be tilted at right angles to the vertical plane of rest of the pendulum by (the small) angle $\epsilon$, the angular movement of the pendulum would be to $e$, not as $mg$ to $mg\sin \theta$, but as $mg$ to $mg\sin (\theta + r)$; that is, the angular movement of the pendulum would be multiplied by +$$\frac{mg}{mg\sin(\theta + r)} = \frac{1}{\sin(\theta + r)}$$ +or $\sin(\theta + r)$; which fraction now represents the sensibility of the instrument. This cannot be increased above $\frac{mg}{mg\sin(\theta)}$ unless by making $mg\sin \theta$ negative, that is, by very slightly inverting the pendulum, so as to speak, as far as regards the action of gravitation upon it; that is, by making the axis of oscillation lean slightly backwards or away from the pendulum, so that it will oscillate under $r - mg\sin(\theta)$ (of course $r$ must be greater than $mg\sin(\theta)$), and its sensibility will be $$\frac{mg}{mg\sin(r - mg\sin(\theta))} = \frac{1}{\sin(r - mg\sin(\theta))}$$ +If $r$ could be absolutely unaffected by viscosity and constant, this would afford a means of increasing the sensibility indefinitely. But the force of torsion or of flexion is interfered with by viscosity, and the present behaviour of a spring depends on its recent history as to temperature and strain. Consequently, when $r - mg\sin(\theta)$ is exceedingly small and the sensibility of the pendulum correspondingly great, the imperfection of the elasticity will become important, and the effect of viscosity will only be for a comparatively short time, that of stable equilibrium. + +If we may surmise from the behaviour of Zöllner's Pendulum on the occasion described in p. 404 of the vol. Phil. Mag., above referred to, the pendulum was thus inverted; though we cannot be sure of this, without knowing the moment of torsion of the watch-springs and the moment of inertia of the pendulum. + +Therefore, since it is obviously desirable that the "Horizontal Pendulum" should be, as nearly as possible, a pure gravitation pendulum, and that its action should depend as little as possible on the force of torsion or of flexion, of the supporting threads, + +ON THE "HORIZONTAL" PENDULUM. + +wire, or bands, these should be as thin as may be practicable; and, to allow of this, they should be made to support the pen- +dulum in such a way as to have as little stress on them as possible. Now, judging from the drawing of Zollner's Pendulum, supposing it to be drawn to scale, the mean stress on each of the watch-springs is at least 3-5 times the weight of the pen- +dulum; they must be strong enough and thick enough to withstand this stress. In wires, therefore, they must have an +unusually great force of torsion to interfere with their use. We +have noted, with the performance of the Pendulum (this ob- +jection is much stronger against Hengelner's Pendulum). To +this we may add that for certain reasons it is desirable that the +gravitation zero and the torsion zero should coincide as nearly +as may be; but when the force of torsion is greater than may +be expected, the coincidence of the two non-coincidence of the zeros is so likewise. (See Note A.) + +It would seem, then, that the mode of suspension illustrated in Fig. 22 is preferable to some of the others now described. The +stress on each thread is less than three fourths of the weight of the pendulum; each thread, therefore, need only be strong enough and thick enough to endure that stress with safety. + +If the edges of a cylinder are smooth and perfectly fine, +nearly, the magnitude of the effect during an angular movement +of the pendulum will evidently be constant, and not, as above, +proportional to the pendulum's angular distance from any +particular point. In this case the effect would not diminish +the sensibility of the instrument; but merely alter very slightly +its zero point, or position of rest. + +In order to avoid any sagging in the support of the +pendulum this should be only not as short as is consistent with other requirements, both solid and strong; and the weight of +the pendulum should be kept down as much as may be, and +stops should be provided to prevent too great departure from +the position of rest. It should of course be contained in a case +or box, proof against movements of the air, and with sufficient +non-conductivity of heat; the inside of the box being lined with + +101 + +102 +ON THE "HORIZONTAL" PENDULUM. + +tin-foil, with sufficient metallic connection with the ground, to guard against unequal distribution of electricity; and the pendulum itself should be made of the least magnetic or (so-called) diamagnetic substance. + +The sensibility of the instrument, or the ratio of the angular movement of the pendulum to the angular tilt to be measured, might, perhaps, be obtained approximately by an exceedingly delicate leveling screw, or by a screw working with a differential motion, but this would not give any very accurate result. A very small lateral tilt in the stand of the pendulum, to be compared with the consequent angular movement of the pendulum. + +But the following would doubtless be a much better way of obtaining the sensibility, if the effect of the resistance of the air were quite negligible. The sensibility is, as we have said, my $g_{\mathrm{air}}$, or the ratio of the forces acting at the centre of mass when the pendulum is hanging freely and when suspended in the manner now in view ; both forces being proportional to the angular distance of the pendulum from its position of rest. But these are inversely proportional to the squares of the corresponding times $t$ and $t_0$, of oscillation. Let the shape of the pendulum be such that its oscillations occur at a frequency $\omega$ obtained from measurements of its parts; this will give the time $t_0$. The time $t$, is known from direct observation; thus $t/t_0^2$, the sensibility of the pendulum, is known. If the resistance of the air be proportional to the velocity of the pendulum in its swing, which it is very approximately for very small velocities, its interference with the isochronism will be exceeded simply by making it slightly increase the period of oscillation, and so make the sensibility calculated in this manner very slightly greater than the truth. + +There seems to be very little likelihood that the moon's tidal force will ever be measured by the Horizontal Pendulum, or by any instrument working as a level does. The experimenter must first make, with Archimedes, the rather important request; + +A diagram showing a horizontal pendulum suspended from a support. + +ON THE "HORIZONTAL" PENDULUM. +103 + +"Give me wherein I may stand." Not to dwell on more- +ments in the earth's crust, disturbances of level by changes of +temperature, the moon's own tidal deformation of the body of +the earth, &c., it would appear that in most cases, at least, a +gentle breeze pressing on the side of a house would make the +whole structure of the house tilt through an angle considerably +greater than the greatest change in the vertical by the moon's +tidal force. + +Note A, from p. 101.—Let $e$ and $e'$, Fig. 25, be two centres of +force varying directly as the distance ; the absolute value of the +forces, or their magnitude at unit distance, being F and F', +Fig. 25. +respectively. It is easily seen that they are equivalent to a +force having the same law, with absolute magnitude F+F', and +with centre C whose distances from $e$ and $e'$ are inversely as F +and F'. (This obtains, of course, not only in the line $ee'$, but +throughout all the space around C.) + +It is evident that if the zeroes in the text do not coincide, and if +the instrument be tilted slightly in the vertical plane of rest of the +pendulum, there will be a horizontal movement of the pendulum, +which might be taken as an indication of a lateral tilt. + +Mess. We are indebted to Mr. Charles Davison, Secretary of +the British Association's Committee on Earth Tremors, for some +information on the subject of this Chapter. + +[ 104 ] + +CHAPTER VII + +THE MOON'S VARIATION. + +This space that can be afforded, in ordinary elementary treatises on astronomy, to the moon's Variation and Parallelistic Inequality is necessarily small; so that various important and interesting matters connected with those lunar inequalities must be left out of consideration. But as our space is at our own disposal, we can afford to give some of these details more of the attention which they deserve. + +We assume that the reader is acquainted with the nature and the general cause of the inequality of the moon's motion called the Variation. It is produced by solar differential forces, tangential and radial, similar to those which produce the tides. Fig. 20 is a diagram of the chief particulars of the Variation one of which is shown in Fig. 21. The letter A represents the moon's motion: E is the position of the earth; ABCD is the moon's orbit round the earth; the moon revolving in the direction indicated by the order of those letters. The sun is supposed to be away to the right, over A, at a distance from O representing 388 times EA, the moon's distance from the earth. The tangential arrows and the radial ones drawn with broken lines show the directions of the tangential and radial directions of the solar disturbing forces, tangential and radial. The contractions will be readily understood: see, mean; get, "greatest"; let, "least"; pl., "place"; vel., "velocity"; g, "gaining"; f, "losing". The terrestrial tangential forces, to be mentioned further on, are not represented from want of room; but this is of no consequence, if it be remembered that in the + +THE MOON'S VARIATION. +105 + +Variation scheme of forces they always agree with the solar tangential forces, both as to reach, or range, and direction. + +The disturbing forces are calculated, *mutatis mutandis*, precisely + +Diagram showing the variation of the moon's velocity due to its distance from the earth. +Fig. 26. + +as the tidal ones. If the earth's mean attraction on the moon be taken as unity, then for a circular lunar orbit, the tangential disturbing force will be $38R^2 \sin 2\theta$, or $\frac{1}{3}E \sin 2\theta$, and the radial force $38R^2 (\cos 2\theta + \frac{1}{3})$; S being the sun's mass, E the earth's mass, D the distance of the sun from the earth, R the distance of the moon from the earth, and $\epsilon$ the moon's elongation from the sun reckoned from the perigee. This last right round to $360^\circ$ is maximum value of the solar tangential force. + +* These expressions show that the Variation forces are, quasi-geometrically, inversely proportional to the cube of the sun's distance. It might seem, at + +106 +THE MOON'S VARIATION. + +which occurs at the octants, is 1/120th of the earth's mean attraction on the moon, that of the radial force outwards at syzygies is 1/30th of the same; its inwardly directed maximum, at quadratures, is half this, or 1/180th. These forces are, then, very small relatively to the earth's attraction on the moon; which circumstance renders it unnecessary to enter into a mathematical discussion of the Variation. Let us note that the tangential forces vanish when sin $2\alpha = 0$; that is at syzygies and quadratures, and the radial ones when cos $2\alpha = \frac{1}{2}$; that is at the four points distant by $\delta^{\circ}44'$ from syzygies. + +The inequalities in the moon's motion, due to these differential forces, are important, for two reasons. In the first place, they present interesting physical and mathematical problems, both dynamical and kinematical; and, in the second place, it is of great moment to know approximately enough their magnitude, for the construction of tables of the moon, by which to be able to predict the moon's true longitude, or angular distance on the ecliptic from the first point of Aries. + +When the latter object is in view, the equation of the moon's angular Variation, or difference between her true and mean longitude, caused by the said forces, is usually given thus— + +$$M' \text{tr.long.} - \text{her m.o.} = C_{\alpha} \sin 2(M' \text{m.o. long.} - \delta) \text{'s.do.},$$ + +(1) + +$$M' \text{being the moon}, N \text{the sun}, \text{and } C \text{the coefficient of the Variation in longitude. (See Note A.)}$$ + +But our present main object is to consider the matter simply on its own account; and as the disturbing forces are connected with the moon's elongation, or angular distance from the sun, it will be simpler and more interesting to consider their effects on this, rather than on her longitude, to which we are now first sight, that they are directly proportional to the cube of R, the moon's distance. But they are proportional only to the first power thereof. The $R^{3}$ comes in on account of the earth's mean attraction on the moon being taken here; for convenience, as unity. + +THE MOON'S VARIATION. +107 + +indifferent. Moreover, we are now concerned with the Variation only in the abstract, or in its purity, as we may express it. We shall therefore take the moon's undisturbed orbit and the sun's straight, or approximately straight, line of sight between both circular, and at the same plane, and the angular velocities of both luminaries in those orbits as constant: so that the sun's true, and mean, longitude will be the same. Subtracting, then, the sun's true longitude from the left side of the above equation, and his (now) equal mean longitude from the other side, and adopting what seems the best value of the coefficient C, we have (from Hansen) the equation + +$$e\cos 43^{\circ}45' \sin 2p;\qquad(2)$$ + +in which e is the moon's true elocation from the sun, and $\varphi$ her mean elongation, for the same instant of time, both reckoned eastwards from the sun to $360^{\circ}$. Thus, then, the moon's pure abstract Variation in elongation, or her departure from $\varphi$, is $+43^{\circ}45'$ sin $2p$. + +Now let $r$ be the moon's actual radius-vector, or distance from to earth, and b her mean distance, and we shall have (from Hansen) + +$$rR(1-\frac{1}{18}\cos 2p).\qquad(3)$$ + +The Variation in the moon's distance from the earth is, then, + +$$-\frac{1}{18}\pi\cos 2p.$$ + +From these equations (2) and (3) in combination may be easily derived a simple geometrical construction for obtaining the moon's position when given any assumption $\varphi_0$ on mean elongation. First let us note that in equation (2), $43^{\circ}45'$ sin $2p$ in equation (2) is, in circular measure, $1/96$. whence the moon's linear departure, forwards or backwards, from the line of her mean radius-vector is $e\cdot r\cdot R\sin 2p$, $g.p.p.$; her departure from her mean distance being, as we have seen, from equation (3) + +$$-\frac{1}{18}\pi\cos 2p.$$ + +108 +THE MOON'S VARIATION. + +In Fig. 27, E is the place of the earth, ES the direction of the sun. The construction is as follows.--Draw Ea to represent, in magnitude and position, the moon's mean radius-vector, at a given time, whose length is 238,820 miles ; SEa is $\varphi$, and $a$ is + +A diagram showing the moon's orbit with respect to the Earth. The diagram includes points labeled M, S, E, and arrows indicating directions. The text describes the construction of this diagram. + +Fig. 27. +M +$\varphi$ +S +Ea + +the moon's mean plane. Draw ab making the angle $2\varphi$ with Ea (or $\varphi$ with ES), to represent the length $1\frac{1}{2}(R_{E}+r_{E})$, or 2003 miles, then draw AM making the angle $4\varphi$ (or $3\varphi$ with ES), to represent the length $1\frac{1}{2}(R_{E}-r_{E})$, or 357 miles; then M is the moon's true place for assumed $\varphi$. (See Note B.) With the exception of the line Ea, which is necessarily vastly too short, this Fig. and Figs. 30 and 31 are drawn to scale; the scale being that of 1 inch = 1000 miles. + +Thus we see that the moon's pure Variation orbit*, according to equations (2) and (3), is a compound epicyclic curve as referred to ES regarded as fixed. Ea is the radius of the deferent circle turning progressively with the moon's mean angular velocity; ab is the radius of the first epicycle turning retrogressively with the same angular velocity; bM is the radius of the second epicycle turning progressively with three times the said angular velocity. + +* By "Variation orbit" we mean the moon's orbit as deformed by the Var. disturbing forces alone; the undisturbed orbit being supposed circular. + +THE MOON'S VARIATION. +109 + +The curve described by the moon about her mean position is, of course, a simple epicyclic, relatively to ES regarded as fixed, whose deferent is the first epicycle above mentioned. It is a four-lobed curve, like that in Fig. 28. Its greatest diameters + +Fig. 28. + +being $\frac{1}{2}R$, and the least $\frac{1}{4}R$. This curve, as its centre is carried round on the end of ES, or R, always keeps the same shortest diameter parallel to ES, so preserving the same section with the plane. The moon describes this curve about her mean place once in a synodical month and retrogressively. When in conjunction she is at $p$ in the curve; when in first octant she is at $q$; when in first quadrature at $r$, &c.* + +The movement of the moon in her Variation orbit can be represented in another manner, which is of considerable interest. It follows directly from the same equations (2) and (3). (See Note C.) + +It is often said simply that the moon's Variation is elon- + +* Of course relatively to fixed space this curve itself rotates one in a year progressively; but we are not now concerned with that. + +A diagram showing a four-lobed curve labeled "V" with points labeled "P", "Q", and "R". The curve is drawn around a central point labeled "E". The lobes are labeled "a", "b", "c", and "d". The lobes are connected by curved lines. + +110 +THE MOON'S VARIATION. + +gation from the sun is proportional to the sine of twice her elongation, that is to sin $2\varphi$. The discrepancy involved is evidently very small. It can be seen without difficulty that if in equation (2) we substitute sin $2\varphi$, it will make a difference in the angular Variation of only --- $\frac{1}{35} \cdot 45^\circ$ sin $4\varphi$. This vanishes at syzygies, quadratures, and octants. It is at its maximum value, alternately negative and positive, at the eight points of the moon's orbit, but this maximum does not amount to $2^\circ$. We may, then, take the liberty of writing equation (2) in the following form, which is more convenient, while always fully accurate enough for our present purpose, and quite accurate at the eight points just mentioned; viz. +$$e \cdot m + \frac{1}{35} \cdot 45^\circ \sin 2\varphi = \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4)$$ + +According to this the moon is at her undisturbed place in elongation at syzygies and at quadratures, most before that place when $45^\circ$ past syzygies, and most behind it when $45^\circ$ past quadratures. + +If we substitute, in equation (3), cos $2\varphi$ for cos $\alpha$, similarly to what we have done with equation (2), this will involve a discrepancy proportional to --- sin $2\varphi$, which varies from zero, at syzygies and quadratures, to its maximum, always negative, at octants; but this maximum is only 1/48th part of the greatest value of the Variation in the moon's radius-vector. When written thus: +$$r = R(1 - y^2)\cos 2\varphi = R(1 - G\cos 2\varphi),$$ +it becomes a polar equation of the moon's Variation orbit; which is quite sufficiently accurate for our present purpose. The pole, of course, is at the earth, and the curve is referred to the line ES as its prime axis, or prime vector; as this revolves once in a year the Variations oval does likewise, along with it. Consequently any inequality whatever in the moon's orbit if subject to no other inequalities than those of the Variation, would be an oval + +THE MOON'S VARIATION. + +with its shortest axis in the line of syzygies, or directed towards the sun, and its longest axis in the line of quadratures; these axes being to each other as $1 - \frac{1}{\sqrt{3}}$ to $1 + \frac{1}{\sqrt{3}}$ or as 67 to 68. The proportion given by Newton was very close to this, viz., 60 to 70. + +Giving to the smallness of the coefficient $\frac{1}{\sqrt{3}}$ equation (5) differs practically but little from that of an ellipse. It gives a curve which is slightly flatter at syzygies and quadratures (where it coincides with (3)) than an ellipse with the same principal axes. The radii of curvature at those points can be easily obtained by the geometrical method; they are, for syzygies, $R_1 = \frac{1 - C'}{1 + C'}$ and for quadratures, $R_2 = \frac{1 + C'}{1 - C'}$; $C'$ being the coefficient $\frac{1}{\sqrt{3}}$. (See Note D.) + +The moon's velocity is greatest at syzygies, least at quadratures, and at its mean at octants. But we shall return to this. + +There are some very interesting particulars, both kinematical and kinetical, connected with the Variation, which, being contrary to what many persons would expect beforehand, present to them, at first sight, the appearance of paradox. + +One of these is that, as we have just seen, the Variation orbit should have its shortest axis directed towards the sun. The dynamics of this proposition are contained in Newton's "Principia", Book III.; but of course the analytical treatment of the question is more powerful and complete. It is most respectfully submitted that the ordinary short popular "proof" of this is quite inadequate, for more than one reason. May we venture, while depreciating the imputation of rashness, to propose another proof, as we hope it to be. In excuse for its length we beg to plead that no proof can be sufficient unless it takes into account all the disturbing causes; -- that is to say, not only the tangential, but also the radial, disturbing forces, as well, and also the condition under which + +$$\frac{d^2x}{dt^2} = \frac{d^2y}{dt^2} = 0$$ + +112 +THE MOON'S VARIATION. +they act, viz., the law of the earth's gravitation attraction on the moon. + +Elementary Proof of the character of the Variation inequalities in Elongation and Radius-Vector.--It is probably impossible to give an entirely priori proof of this which shall be both simple and quite complete. Some of the following sketch-argument are founded for a study of the variation inequalities of the perturbing forces and of the perturbations with which we are now engaged. We know, a priori (see p. 106), that, relatively to the earth's attraction on the moon, the Variation perturbing forces are very small; and we know from observation that the resulting perturbations are so likewise. We can, then, consider the actions of the tangential and of the radial forces separately, and see how they affect the moon's orbit and its orbital effects; these being comparable to different sets of "small oscillations." Supposing still, for simplicity, the moon's undisturbed orbit and the sun's relative orbit round the earth to be both circular, since we know, a priori, that the scheme of the Variation disturbing forces is symmetrical on each side of the line of syzygies and also on each side of that of quadratures, we may conclude that, if such a scheme holds, we are justified in concluding that that orbit must be itself symmetrical on both sides of each of those lines, as principal axes. + +In Fig. 29 the circle represents the moon's undisturbed orbit supposed circular; the sun being over A, and the moon revolving in the direction ABCD. + +We shall neglect, for the present, the sun's relative annual oscillations about its mean orbit; the result of which, as we shall see, is merely to increase the effects now to be considered. + +We shall take first the solar tangential disturbing forces considered by themselves, and as acting on an originally circular orbit of the moon round the earth. We shall first suppose the sun's disturbing power to begin to exist when the moon is passing D. We are, however, in a little difficulty here. The + +THE MOON'S VARIATION. + +113 + +sun never began to act at any particular point on a previously undisturbed lunar orbit. If we begin at D with force $a$, as we shall call it, we must compound its effect with that of its successor, which we have marked $b$; but we have no more right to do this than to compound its effect with that of its predecessor $d$. + +Fig. 29. + +A diagram showing the positions of points A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. + +In order to approximate to the right thing, we must take different starting-points in succession, and then combine the results. It will be simplest to take the four points of syzygy and of quadrature, in succession, as starting-points. We shall then find that the tangential force $a$, or the couple the impulse of $a$, or the impulse of its product with its time of acting, with which we have to do. Now the force $a$ is a maximum at the octant, which is the middle of its reach, or range, DA, and its magnitude, which is, as we know, very small even at the maximum, is always, as we see from its formula, p. 105, equal at equal distances on each side of its maximum, and zero at both ends of its range. Hence the impulse of $a$ will be zero for some length towards each end of its reach. It will, therefore, involve a very small inaccuracy, as relates to our present subject, if we regard the whole impulse as condensed into a very short-lived impulse of the same magnitude, like an impact, acting tangentially at the octant. We can treat similarly the opposing + +1 + +114 +THE MOON'S VARIATION. + +tangential impulse $b$. Now, in consequence of the law of the earth's attraction, which would make the moon move in a focal ellipse round the earth, the impulse $a$, by itself, would produce an elliptical lunar orbit with an apogee 180° distant, at the octant under the letter $c$. But the opposing tangential impulse $b$, which is equal to $a$, would, if it acted by itself on the still undisturbed moon, cause it to describe an ellipse with its apogee very near the earth, but under the letter $d$. These two ellipses would have a very small proportional difference as to magnitude; although the former would be entirely outside, and the latter entirely inside, the original circular orbit. The eccentricities of the two ellipses would be very nearly the same. Now these two apogees under $b$ and $a$, only 90° apart, would combine or coalesce, as is evident, into an apogee very nearly half way between them. This apogee would be nearer to the earth than any other, and this apogee would be above the circle in the figure; as the former ellipse would cross the line of EB at a height above the circle due, inter alia, to the distance of three octants from the apse under the letter $c$; while the other ellipse would cross EB below at a depth due to the distance of only one octant from its apex under $a$. The composition of these greater and smaller fall would give us an apogee of EB above the circle. Thus the impulses $a$ and $b$ acting together, apart from the others, would produce an apogee very near B above the circle. Now let us start at A with impulse $a$. We shall find, in a corresponding manner, that impulses $b$ and $c$, acting together apart from any others, would produce a perigee very close to C, and below the circle; and that when they act together with impulse $d$, they produce D above the circle; and $d$ and $c$ together a perigee near A below the circle. But we must not use each tangential impulse twice over; therefore, to avoid this, we must take only half of each in our successive stages round the lunar orbit. Thus the tangential disturbing forces alone would deform the moon's originally circular orbit into an oval with its major axis in quadrature". + +We see here that though the tendency of the immediate local action of the tangential force in the quadrant DA would be to make the moon rise + +THE MOON'S VARIATION. +115 + +This deformation of the orbit by the tangential disturbing forces gives rise, of course, to tangential components of the earth's attraction on the moon, or, as we shall call them, terres- trial tangential forces, whose positions and directions are, in the case of the Variation, the same as the solar ones. These, there- fore, are the causes of the moon's variation in her mean rate of revolution round the earth. Thus the solar tangential forces are the means of causing considerably greater inequalities in the moon's velocity, both linear and angular, than they could produce by their immediate local action. + +Therefore, if the tangential disturbing forces were to act by themselves, apart from the radial ones, the moon's linear velocity would be constant at all times, and her angular velocity would be so, a fortiori, as regards her angular velocity. But we cannot assume, at once, that these things must be actually so; for these forces do not act by themselves. If it should so happen that the radial forces by themselves would produce a sufficiently greater elongation of the orbit in syzygies, the terrestrial tan- gential forces created thereby, which would then be oppositely directed to the solar one, might be the greater of the two; so that the moon's velocity might be least at syzygies, and greatest at quadratures. + +We therefore turn now to the radial disturbing forces; first taking by themselves the outwardly directed ones, whose action extends for 64° each on either side of both syzygies. It will be seen, on a very little consideration, that these by themselves would cause a considerable elongation of the orbit at every angular distance after syzygy; because their effect in increasing the moon's distance from the earth must evidently continue for some time after they have ceased to act with their greatest efficiency, which happens at syzygy. We shall see in Note Z that in + +from the earth, yet in consequence of the whole general action even of the tangential disturbing forces alone, the moon is really falling earthward all through that quadrant. This illustrates a principle to which we shall refer again. + +Z2 + +116 +THE MOON'S VARIATION. + +consequence of the law of the earth's attraction, the greatest lengthening, or apogee, will be exceedingly near to quadrature. Similarly, of course, with the (outward) radial forces about opposition. + +Now the inwardly directed radial forces, extending for 35° 16' on each side of both quadratures, would, by themselves, tend to produce perigees very near to both syzygies. But the latter force, being almost equal to the former, and since these forces are roughly of only one third the value of the outward impulses; since the average magnitude of the force is only about one half, and their time of acting only about two thirds, those of the former (see Note F) and therefore the result of their action is much smaller than that of the outwardly directed forces; but it is auxiliary as regards the general effect now in question. This point, however, is not so important as that which concerns us with respect to its existence on the law of the earth's attraction, is controlled thereby, so that it cannot exceed a certain magnitude. Thus the radial disturbing forces, by themselves, would produce a deformation of the moon's orbit similar in general character to that due to the tangential ones; and therefore would by themselves give rise to tangential components of the earth's attraction on the moon, consisting in inequalities in the moon's velocity, both linear and angular, similar to those produced by the tangential forces (but much smaller).* + +We now know that we actually have maximum linear velocities of the moon at the ends of the (shorter) syzygy axis, and minimum velocities at the ends of the (longer) quadrature axis; and that this is true for both of the angular velocities. +* We see that we must not institute too close a comparison between the variations of velocity and those of attraction along different axes. Though the system of the lunar differential forces producing the tides and that of the solar ones producing the Variation are precisely similar, yet they are acting under very different conditions. Though the Variation oral is somewhat less than that of the tide-producing forces, yet still it is greater than the tidal one is so placed only under the special circumstance that the water is of less than a certain depth. + +THE MOON'S VARIATION. +117 + +There are, therefore, two reasons for expecting that the Variation orbit must be flatter at syzygies, and more curved at quadratures, than elsewhere. But probably this cannot be proved by any simple considerations such as those to which we are now accustomed; but it may turn out that various particular circumstances connected with the Variation will turn out contrary to what a large proportion of reasonable persons would anticipate, this might be one of them ; and it is, relatively to what we know so far, quite possible that this natural expectation might prove erroneous. Since the moon is nearest to the earth at syzygies, it might very well happen that the consequent effect of the earth's attraction on the moon's orbit would be a greater disturbing force at those points (we shall see indeed further on that this is actually so), and therefore the possibly greater earthward force at syzygies might overcome the effect of the moon's greater velocity at those points, as regards the curvature of the orbit there; and correspondingly at quadratures. This much, however, is quite clear; viz., that there are less curves at syzygies than at the quadrilateral circle there, and this is observed at quadratures than the greater equivalent circle there. But this will not prove the matter in question; because, for all we could yet say to the contrary, the orbit might be four-leaved. + +We have already alluded to the fact that the relative annual revolution of the sun round the earth increases the lunar longitude by 360° every 27 years; and this gives rise to the system of alternating disturbing forces a longer period, viz., half a synodical month, than what they would have without it, viz., half a sidereal month. + +The disturbing forces, then, have the effects now mentioned in consequence of the condition that the earth's attraction on the moon is inversely proportional to the square of the distance; so that that attraction is always endeavouring to make the dis- turbed lunar orbit an ellipse with the earth in one focus. But this same condition, which determines the character of those + +118 +THE MOON'S VARIATION. + +effects, determines also the limit of their magnitude under the action of those disturbing forces. The earth's attraction, because of the law of its action, keeps down the height of the apogee at each quadrature by endeavouring to make there a perigee answering to the apogee of the preceding quadrature. The inclination of the plane of the orbit to that of the revolving forces, and the focal ellipse, that the earth's attraction is always trying to produce in the moon's disturbed orbit, afford the earth an opportunity and power of retraction against the deformation, which would be greater than it is if the deformation were so. The sensibly constant disturbing forces are able to produce, by accumulation, only that amount of deformation at which further increase would become intolerable, and at which the earth acquires sufficient power of retraction and control to balance the action of those forces. + +The opposition, in this respect, between the solar and the terrestrial forces is due to the fact that the scheme of disturbing forces and the consequent Var. orbit are symmetrical on each side of two rectangular arcs passing through the earth's foci. This symmetry is destroyed by the fact, which the earth's attraction is always endeavouring to produce, would have only one axis of symmetry passing through the earth. There is, in this respect, an important and interesting difference between the Variation and the Parallactic Inequality, to which we shall return in the next chapter. (See Note G.) + +We come now to another apparent paradox, already alluded to, connected with the Variation, which it is particularly necessary to notice as it is so generally overlooked, sometimes with inconvenient results. It is this, that the effect of the terrestrial tangential force in producing the moon's Variation in longitude is considerably greater than the direct effect of the solar tangential force. + +We know already that, the earth's mean attraction on the moon being taken as unity, the solar tangential force is $\frac{1}{2}g\sin 2\varphi$; + +THE MOON'S VARIATION. +119 + +but, accepting equation (4) as the polar equation of the Var. orbit, which it is, $\varphi_{\text{moon}}$ $p_{\text{moon}}$, it is not difficult to find that the terrestrial tangential force in $v_{t}^{2}$ sin 2$\theta$ $(1+\frac{1}{r}\cos 2\theta)$, $\varphi_{\text{earth}}$ $p_{\text{earth}}$. (See Note II.) Therefore the two forces, practically speaking, vary very nearly according to the same law, viz., as on page 306, where we found that at any given point of the Variation orbit, in a ratio never less than 120 to 68; or say 7 to 4, very nearly. But the shares of the moon's whole displacement in elongation produced by these two forces at any given point are very approximately proportional to the respective magnitudes of the forces *). Those shares are, therefore, to each other very nearly in the said proportion of 7 to 4. + +Thus we see that the inequalities of the moon's velocity in various places much more by means of their deformation of the lunar orbit than by their direct immediate influence on the moon's velocity near those places. + +If the general action of the solar disturbing forces had, by accumulation of effects, changed the assumed originally circular orbit into its present shape, but with the longest axis directed to the sun, as it might have done, for all that we could tell beforehand; the contrary, and as most people would expect it to do, the solar tangential forces, while still very nearly indeed of their present magnitude, would retain, of course, their present direction. The terrestrial tangential force, which still almost precisely of their present magnitude, would be reversed in direction. The terrestrial would actually overpower the solar tangential forces, as regards their immediate effect on the moon's velocity; and the result would be that notwithstanding the acceleration due to the solar tangential forces, the moon would go gradually slower in the quadrant DA; and, for a + +*) This is so; but only because the two forces vary so very nearly according to the same law, and because the changes of velocity due to each of the forces are so exceedingly small relatively to the mean velocity of the moon in her orbit round the earth. + +120 +THE MOON'S VARIATION. + +corresponding reason, gradually faster in the quadrant AB; and so on. +It must not be thought an absurdity to contemplate before-hand the possibility of such an action ever taking place; for we shall find an instance of it farther on in the Parallactic In- spirations. +Therefore we cannot lay down that the moon must be necessarily quickening or slackening her pace, according as the solar tangential forces are acting with or against her motion, until we first know enough respecting the deformation of the Variation orbit and the position of its greatest and least axes. +If the attention be fixed too strongly on the immediate, direct act of the tangential disturbing forces, it will tend naturally to the over-statement frequently met with, viz., that the Var. in longitude is almost entirely due to the tangential disturbing forces. This would undoubtedly be so if the Var. depended principally on the direct action of the two disturbing forces; but we have seen that such is by no means the case. +The Var. in longitude is, therefore, principally due to the action of the radial disturbing forces, which produce a deformation of the Var. in longitude greater than that of the tangential forces. The tangential disturbing forces are, indeed, more important than the radial ones in causing the Var. in longitude; but the share of that inequality is assigned to them is not as much as double that of their effect upon the Var. in latitude. +To return to a matter alluded to above — It might be supposed that since the radial disturbing force, which is directed away from the earth at syzygies, is at its maximum at those points, therefore the whole earthward pull on the moon is least at those points, and, for a corresponding reason, greatest at quadratures. +Ary. in *Grauntation*, p. 69, shows that he was aware of the enormity of this error; but unfortunately he did not find this defect until he had delivered lectures on astronomy (which (entitled *Pygmaea Astronomy*), he had forgotten his own know- + +A page from a book. + +THE MOON'S VARIATION. +121 + +ledge of its incorrectness. We can easily see for ourselves how the matter stands. The earth's mean attraction on the moon being taken as unity, the solar radial force at syzygies is, as we have seen, 1/60, which is to be deducted from the earth's attraction at syzygy; but, on the other hand, since the moon's distance from the sun at syzygies is 1/300th that at quadratures by 1/150th, the earth's attraction on the moon, which varies inversely as the square of the distance, is at those points greater than the mean by 1/30th, or 1/60th. Therefore the whole earthward pull on the moon at syzygies is $1 - \frac{1}{60} + \frac{1}{300}$, which is greater than unity, the mean, and (as we can easily see) a maximum. Similarly the whole earthward pull on the moon at quadratures is $1 - \frac{1}{60} - \frac{1}{300}$, which is less than unity, the mean, and (as we can easily see) a minimum. + +The Var. forces being proportional to the inverse cube of the sun's distance, it might seem reasonable to believe that the moon's Var. in elongation must be proportional to the same. But in reality this lunar inequality is a very complex function of that distance, which would vary, not indeed very differently from the inverse cube thereof, but at a higher rate. + +We may here refer to what some might regard, at first sight, as a kinematical paradox; though it does not belong specially to the Variation. Since the moon is being retarded, both by the solar and the terrestrial tangential forces, while passing from A to B, it might be thought that she must be behind her mean place in order to gain time in her motion towards B; while she enters on that quadrant at her mean place, she is then moving with her greatest velocity; and as long as her velocity is above the mean, she is gaining on her mean place which she had at A; though her velocity be in the act of diminishing down to the mean under both the opposing tangential forces. Similarly this quadrant of retardation is being accentuated there by both the essential and accidental forces; because when entering on that quadrant at her mean place, she + +129 +THE MOON'S VARIATION. + +is moving with her least velocity; and she must continue to lose on her mean place until her velocity has been increased by the tangential forces up to the mean. When at her mean place, as at syzygies and quadratures, she is moving with greatest or least velocity, respectively; because she has then been, for the longest time, either gaining or losing on her mean place. When most before or behind her mean place, at the octants, she is moving with mean velocity; because she has then just ceased to gain or to lose, respectively, on her mean place. + +Taking these considerations in connection with the results of an inspection of equation (4), we see that we can fill in, for ourselves, all the writing in Fig. 20, the diagram of the Variation. + +Note A, from p. 106.—It is evident that equation (1), by itself, cannot give the accurate value of the effects of the Var. forces whose magnitudes always depend on the moon's true elongation from the sun, or the difference between the moon's and the sun's true longitudes. For it is intended only as a first step towards obtaining the moon's Variation; and this Variation would be supplemented by other more smaller equations of the moon's motions connected with this inequality. The angular distance described between the brackets in equation (1) has been given in several other ways; e.g., as the moon's equated long., misses the sun's true long., as the moon's mean long., misses the sun's true long., etc.; but it is not correct to say that they are only from these, or very slightly from each other. All the differences are small or more or less completely made up for by subsidiary equations. It is only very approximately correct to say, as is often said, speaking roughly, that the Variation vanishes at syzygies and quadratures; this would be true only if the angular distance within the brackets were the moon's true, misses the sun's true, longitude. + +Note B, from p. 108.—This can be seen as follows. (In Fig. 30, the points marked a, b, M are the same as those similarly marked in Fig. 27; and neglecting the line Ee, the + +THE MOON'S VARIATION. +123 + +scale is the same in both Figs.) Eo being the moon's mean radius-vector, necessarily drawn vastly too short relatively to the other lines, draw ae making the angle 2p with Ee, and of length to represent $\frac{dM}{dR}$; then draw ac parallel to Ee and sensibly pointing backwards to the earth, draw ad at right angles to Ee and ac; then let $ae = \frac{dM}{dR} \sin 2p$. Take e so that $ae$ may Fig. 30. + +$$\begin{array}{c} +\text{E} \\ +\text{a} \\ +\text{b} \\ +\text{c} \\ +\text{d} \\ +\text{e} \\ +\end{array}$$ + +represent $1_{2}^{1}$R, draw em parallel to ad; then $dM = \frac{1}{2}R \cos 2p$, and M is the moon's true place. Now oe = $(\frac{dM}{dR})^{-1}R$. Bisect it in b, and draw bm. Then be and bm are both $(\frac{dM}{dR})^{-1}R$, and ab is $\frac{1}{2}R + (\frac{dM}{dR})^{-1}R$, or $(\frac{1}{2}R + \frac{1}{2}R)$. The angle $Mbe = 2oca = 2Eac = 4p$; whence the statement in text follows. + +Note C, from p. 109.—The other manner is as follows. (In Fig. 31 the points marked a, c, M are the same as those similarly marked in Fig. 30, the scale being still the same.) E is the place of the earth, and ES points to the moon. Let Es be R, the moon's mean radius-vector rotating uniformly with the moon's mean angular motion in elongation; a is then the moon's mean position in space. With centre e and radius $\frac{1}{2}R$, describe a circle, as shown in the diagram. Draw the radius ae, making the angle $2p$ with Es; then ae = $\frac{1}{2}R \sin 2p$. Now produce fe and let fM be to fe in the proportion of the two Var. coefficients, $\frac{dM}{dR}$ and $\frac{1}{2}x$ (very) + +124 +THE MOON'S VARIATION. + +Fig. 31. + +A diagram showing the Moon's position relative to Earth and Sun. The Moon is shown at various positions around its orbit, with lines connecting it to the Earth and Sun. The diagram includes labels for the Moon (M), Earth (E), and Sun (S). A line extends from the Earth to the Sun, labeled "d to S". Another line extends from the Moon to the Sun, labeled "m to S". The diagram also includes a label "F" near the top of the Moon. + +10 + +THE MOON'S VARIATION. + +nearly as 7 to 5); then $M$ is the position of the moon, for her assumed mean elongation $\varphi$; and of $a + \frac{1}{2}R\cos 2\varphi$, (since $E$ differs only insensibly from EM) the change in length of the moon's radius-vector, for assumed $\varphi$. The point $M$ describes round $a$, as centre, an ellipse which is as though it were rigidly attached to the line $EA$, and therefore rotates about its centre once in a month progressively, relatively to ES regarded as fixed, so that its semi-major axis is always perpendicular to the direction of rotation round the earth; its semi-axes major and minor $\alpha$ and $\beta$ being $\frac{1}{2}R\cos \varphi$ and $\frac{1}{2}R\sin \varphi$, respectively; and to each other in the proportion of the two Var. coefficients. As $a$ rotates round $a$, relatively to $a_0$, with twice the moon's mean angular velocity in elongation, $M$ describes the whole ellipse in half a synodical month; and the motion therein being retrograde, or in a direction contrary to that of the sun's apparent motion round the earth. At the time of both syzygies the moon is at $h$ in the ellipse, and nearest the earth; and at the times of both quadratures she is at $a$, farthest from the earth; and when in octants she is at $\gamma$, or at $a$, and at her mean distance. This is very approximately so; but only because the semi-axis-major of the ellipse is so small relatively to the semi-axis-minor of the ellipse. The ellipse has necessarily been drawn in the diagram exactly too large in proportion to $EA$, the moon's mean distance. + +It will be observed that the components of $M$'s motion parallel to $hk_1$ and to $gk_2$, are simple harmonic motions, and that the moon describes the ellipse, relatively to its (rotating) principal axes, as she would be stationary under the action of a force varying inversely as the square of her distance; therefore describing the rotating ellipse with a constant aperture velocity. + +The very approximate correctness of this is due to the fact that the dimensions of the ellipse are so small relatively to the moon's mean radius-vector $EA$. + +To obtain a graphical representation of the solar disturbing forces, we return to the expression for the radial force, viz., + +$$3SR^2(\cos 2\varphi + \frac{1}{2})$$ + +and to that for the tangential force, + +126 +THE MOON'S VARIATION. + +viz., $3\mathrm{SR}^2 = 2\mathrm{ED} \sin 2x$ ; the earth's attraction on the moon being unity. +Let us substitute in these $\varrho$ for $e$, which will involve an exceedingly small inequality. Then, to use Fig. 31 for a different purpose, if we take the circle of radius $r$ to represent the coefficient of these expressions, and if we use one third of said radius, $fm$ will represent, on the same scale, the radial disturbing force for assumed $\varphi$, and $of$ the tangential disturbing force, and $em$ will represent, $qam$ proor, both in magnitude, direction, and sense (out of course not in position), the whole disturbing force acting on M. Its magnitude is said radius of the circle $x\sqrt{1+6\cos 2p}$; and its inclination to E2 is +$$\tan^{-1}\cos 2p + j$$ + +Note D, from p. 111.—This can be seen as follows from equation (5). It and $O$ being as in text, let $c$ be the radius of curvature at the points in question, and $a$ an indefinitely small elongation of the moon, for which equations (5) and (6) coincide. +Now $p = a + c$ (from tangent). But, for sines, +$$\sin^2 a = R(1 - C)\sin e$$ +and +$$2\text{ fall } m = \frac{R(1-C)}{\cos e} - \frac{R(1+C)}{\cos e}$$ +$$= \frac{2R}{\cos e}\left(1 - C - \left(1 - C\right)\cos^2 e\right)$$ + +By the addition and subtraction of $\cos e$ within the large parentheses this becomes +$$\frac{2R}{\cos e}\left((1-C)(1-\cos e) - 2C(1-\cos^2 e)\cos e\right)$$ +$$= \frac{R\cos e}{2}\left(\frac{(1-C)(1-\cos^2 e)}{1 - (1-C)(1-\cos e)} - \frac{2C(1-\cos^2 e)}{1 - (1-C)(1-\cos e)}\right)$$ + +Dividing above and below by $1 - \cos e$, and then making $e = 0$, we obtain the result in text. + +THE MOON'S VARIATION. +127 + +In an ellipse whose semi-axes major and minor are $a$ and $b$, respectively, the radius of curvature at the apse is $\frac{ab}{a^2}$, and that at the ends of the axis-minor $\frac{ab}{a^2}$. Therefore if the Var. orbit were an ellipse with the same principal axes, $p$ would be at syzygies $R(1+\frac{C'}{C})$, and at quadratures $R(1-\frac{C'}{C})$. Taking $R$ as unity, and $C$ as 0.00738, we find the following values for $p$ ---: + + + + + + + + + + + + + + + + +
Var.at apseat centreat apseat centre
In ellipse0-02300-07580-02300-0758
+ +which verifies the anticipation in text that the Var. orbit is very slightly flatter than an ellipse, with the same principal axes, both at syzygies and at quadratures. + +Now E, from p. 115.--This will be sufficiently seen from the following Table. It is a well-known principle (containing your attention now to the observer) that if a body be projected from a given point in presence of a given centre of attraction he has only the law of gravitation, with a given velocity not too great for the description of an ellipse about that centre, the ellipse described by the body will have the same axis-major, whatever be the direction of discharge. + +Now let us consider at first describing a circular orbit abd, Fig. 32, with radius $r$ about the centre of force $F$, in the direction of the arrow. At the point $a$ the direction of the body's motion is changed outwardly, say by the angle $\theta$; and it proceeds to describe an ellipse $ge$, of which a focus is at $c$, and whose semi-axis-major is equal to $r$, the radius of the circle. Since $ce$, drawn from the focus, is equal to the semi-axis-major of the ellipse, it is at the end of the axis-minor thereof. Consequently its angular distance from $a$ is equal to $\theta$. The motion at $a$ gives the direction of the axis-minor of all those apices of which we are in quest. The geocentric angular distance of the apsege from $a$ is $90^\circ - \theta$; and $f$ being the centre + +128 +THE MOON'S VARIATION. + +of the ellipse the height of the apogee above the circle is equal to $c$, or $\sin \alpha$. The same Fig. can be used (by supposing the body to be describing the circle in the opposite direction) to show that if the change of the direction of the body's motion at $a$ had been + +Fig. 32. + +downwards, and of magnitude $\theta$, the resulting perigee at $p$ would be $90^\circ + \theta$ from $a$, and fall below the circle at that point, $r\sin \theta$. + +Therefore, whether the deflection at $a$ be upwards or downwards, the new orbit will be an ellipse whose axis-major is $2r$, and axis-minor $2r\cos \theta$; and if $\theta$ be very small, the apes are distant from $a$ by $90^\circ$ very little. + +We need not go any further, because the actual conditions are slightly different from what we have just considered. The moon is not simply deflected outwards without change of velocity by the radial forces near syzygies; though the condition nearly approaches this, as the radial disturbing forces are so very small. But the difference of conditions is evidently in favour of an apogee both higher and nearer to quadrature than what we have been contemplating. The deflection does + +THE MOON'S VARIATION. +129 + +not indeed occur at a single point; but it may be regarded as the result of a large number of exceedingly small, outwardly directed, radial impulses, whose magnitudes are very nearly equal at equal distances on each side of syzygy. The whole effect is, therefore, different from that of a single impulse at syzygy equal to the sum of the others; but, as regards our present purpose, the difference is quite unimportant. + +Note F, from p. 116.—We have seen, p. 105, that the radial disturbing force vanishes at the four points of the moon's orbit distant $5^{\circ}44$ from syzygy, marked OOOO in Fig. 33. The arc BO is slightly less than two thirds of AO; and, the changes of the moon's velocity being small, her times of describing OB + +A diagram showing the positions of O, B, A, C, D, and S. The circle represents the moon's orbit around the Earth. The line AB is drawn from B to A. The line OC is drawn from C to O. The line OD is drawn from D to O. The line OS is drawn from S to O. The line OA is drawn from A to O. The line BC is drawn from B to C. The line CD is drawn from C to D. The line DS is drawn from D to S. +Fig. 33. + +and AO are still more nearly in the same ratio. Again, we have seen above that the inward radial force at B is half the outward radial force at A; and therefore, as it is easy to see, the average inward force on each side of quadrature is somewhat less than half that on each side of syzygy. + +In consequence of the outwardly exceeding so much the inwardly directed radial impulses, the earthward pull on the moon is, on the whole, diminished; and therefore the mean distance of the moon from the earth is by them increased. + +K + +130 +THE MOON'S VARIATION. + +Note G, from p. 118.—It will be found from simple considera- +tions similar to the above, mutatis mutandis, that if the gravita- +tion attraction varied, not according to its actual law, but +according to that other law of force so frequent in nature, viz., +directly as the distance, the solar disturbing forces, if turned on +to the moon's orbit, would be continually increasing, until that +orbit into what might be called an "ellipse," with the earth at +its centre, whose axis-major, in the line of the second and fourth +octants, would be continually increasing, whilst its axis-minor +would be continually decreasing (more rapidly), until the moon +came into collision with the earth. The instantaneous ellipse +would be always changing; and finally, when the moon came +in bisecting this line, there would be a baffling action +between the solar and the terrestrial forces. The solar tangential +forces would have the same positions, relatively to the sun, as +they have now; though, of course, their directions would be reversed. +The earth's attraction, owing to its now supposed law, +would make the lunar orbit, when disturbed, a central +ellipse; and consequently the solar disturbing forces would fall in with this and go on increasing the ellipticity. The radial disturbing forces, always directed inwards, would be proportional to the moon's radius-vector, like +the earth's attraction, and would therefore conspire therewith. +There would be, moreover, this seemingly curious result, that, +regardless of any variation in the law of gravitation, in com- +parison with that of the moon, the Var, forsooth would not sensibly alter with the sun's distance; instead of being, as they actually are, +inversely proportional to the cube of that distance. + +Note II. from p. 119.—This will be seen thus. In Fig. 34, E +is the earth's place; $a$ is the moon's radius-vector for the point $a$, In +question; in the Var., orbit, of which the curve $a$ represents +portion, and $de$ indefinitely small alteration of $\epsilon$, the moon's elongation. Let $b$ be the angle between the radius-vector and +the curve at $a$, or the tangent thereto. Then, $B$, the moon's + +THE MOON'S VARIATION. +131 + +mean radius-vector, being taken as unity, we have +$$\frac{dr}{dt} = 1 - 2 \cos 2\theta$$ (p. 110) and +$$\frac{d^2r}{dt^2} = 2C \sin 2\theta.$$ + +The earth's attraction at the point in question is $\frac{1}{r^3}$, the mean attraction being unity. But +$$\frac{1}{r^3} = 1 + 2C \cos 2\theta,$$ quæn præx. + +This multiplied by $\cos \theta$ is the terrestrial tangential force. + +Fig. 54. + +Now $\cos \theta = \cot \theta$, $q$, $pr$; as $\theta$ differs so very slightly from a right angle. Draw $cb$ perpendicular to $Ea$. Then $\cos \theta = \frac{dc}{bc} = \frac{dr}{dt} = \frac{2C \sin 2\theta}{(1 - C \cos 2\theta)} = 2C \sin 2\theta(1 + C \cos 2\theta), q$, $pr$. + +Therefore the terrestrial tangential force is +$$= (1 + 2C \cos 2\theta)2C \sin 2\theta(1 + C \cos 2\theta)$$ +$$= 2C \sin 2\theta(1 + 3C \cos 2\theta), q$, $pr$. +$$= \frac{1}{r^3}(2C(1 + C \cos 2\theta).$$ + +Q. E. D. + +This is never less than $\frac{1}{r^3}\sin 2\theta$; while the solar tangential force is $\frac{1}{r^3}\sin 2\theta$ (p. 105). Therefore the proportion of the terrestrial to the solar tangential force, at any point in the lunar Variation orbit, is always at least as high as $120$ to $68$, or as $7$ to $4$, very nearly. + +$x_2$ + +[ 132 ] + +CHAPTER VIII. +THE MOON'S PARALLACTIC INEQUALITY. + +We now turn to the moon's Parallactic Inequality, whose scheme of solar disturbing forces and of changes of velocity, &c., are indicated in Fig. 35. + +In the Variation scheme the disturbing forces, both tangential and radial, on the sunward side of the moon's orbit and those on the opposite side are represented equal in amount; however, they evidently do not form a mean slightly greater than the latter slightly less, than the mean. To remedy this we must now add to those on the sunward side the necessary differential forces having the same direction; and we must subtract from the Var. forces, both radial and tangential, on the off side of the orbit, the same differential forces; or, in other words, join with them the same increments of velocity in the contrary directions. + +The constituents of P.I. forces, with which we have to do, The inwardly directed radial disturbing forces at B and D, in the Variation orbit, Fig. 26, are not affected by the difference between the sunward and the other side of the Var. orbit, and we have put no arrows at those places in Fig. 35. The sun being over A, these are arranged with their heads upwards. The outwardly tangential forces, as mentioned presently, are omitted to avoid confusion. They are directed oppositely to the solar ones; they are, however, at their maximum at both quadratures, while the solar ones vanish at those points. All vanish at syzygies. + +THE MOON'S PARALLACTIC INEQUALITY. + +The P.I. forces, being second differences, or differences between what were themselves only differential forces, are exceedingly small. It is easy to find that, taking the earth's mean attraction on the moon as unity, the solar P.I. tangential force = $\frac{SR^2}{ED^2}$ × (sin ε - sin λ), and that the radial force = $\frac{SR^2}{ED^2}(\cos^2\epsilon - \frac{1}{2}\cos 2\epsilon)$; all the letters here having the same meaning as they have in the expressions for the Variation forces in p. 105. (See Noz A.) The + +Fig. 35. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. +no figuring on mean pl. + +Fig. 35 + +greatest P.I. forces are the radial ones at syzygies; and these are only $1/23,300$th of the earth's mean attraction on the moon. They are about $1/250$th of the Var. radial forces at the same points; these two sets of forces are mutually cancelled and considered by themselves. Since these two sets of disturbances are both very small, they can be combined like two sets + +134 +THE MOON'S PARALLACTIC INEQUALITY. + +of "small oscillations" by simple superposition. We shall, therefore, as we did with the Var., neglect now all other deformations of the moon's orbit and inequalities in her motion, and suppose the moon's undisturbed orbit and the sun's relative annual orbit round the earth to be both circular, and the angular velocities therein uniform, and the P.L. forces to be the only disturbing forces acting on the moon. The lunar orbit round the earth resulting from this will be a circle of radius $r$, let $a$ be the semi-axis of elongation, as above in the pure P.L. orbit, and her mean elongation, or that in the undisturbed circular orbit, both reckoned eastwards up to 360°; the sun's angular motion of apparent revolution round the earth being supposed, as in Chapt. VII., constant, for simplicity. Then we have + +$$e = \cos^{-1} \frac{1}{\sqrt{1 + 5 \cos^2 \varphi}} \cdot \cdot \cdot (1)$$ + +The moon's Parallactic Inequality in elongation is, then, + +$$-2^{\circ} 5' \sin \varphi.$$ We have adopted the coefficient $2^{\circ} 5'$ from the latest investigations of the American astronomers; Hansen gives a smaller value for it. + +Observations may be made on this equation (1) corresponding to those in Note A of the preceding chapter on the Variation ; but they are probably unnecessary. + +Let $r$ be the moon's actual, and $R$ her mean, radius-vector, or distance from the earth. Then we have + +$$r = R(1 - \frac{1}{\sqrt{1 + 5 \cos^2 \varphi}}) \cdot \cdot \cdot (2)$$ + +The Moon's P.L. in radius-vector is, then, + +$$\frac{R}{\sqrt{1 + 5 \cos^2 \varphi}}.$$ + +From these two equations, in combination, may be derived a simple geometrical construction for obtaining the moon's position in space, for any assumed $\varphi$, or mean elongation. First: let us note that the coefficient $2^{\circ} 5'$, in equation (1), is, in circular measure, + +$$\frac{720}{360}$$ + +whence the moon's linear departure, backwards or forwards, from the line of her mean radius-vector is + +THE MOON'S PARALLACTIC INEQUALITY. + +R +$$\frac{1}{1650} \sin \varphi, q, pr;$$ her departure from her mean distance from the earth being, as we have seen, $$\pm \frac{R}{(350)} \cos \varphi.$$ + +In Fig. 36, E is the place of the earth, and ES the direction of the sun. The construction is as follows:---Draw EO to represent, in magnitude and position, the mean radius-vector, at some given time, whose length is 238,820 miles; SEo is $q$, and $\alpha$ is the moon's mean position. Draw $ab$ sunward, making the angle $\varphi$ with the line of Eq (that is parallel to ES), and of magnitude to represent $$\frac{1}{(1650)} \sin \varphi,$$ or 105 miles; then draw $bM$ making the angle $2p$ with $ab$ (and with BS), and of length to represent $$\frac{1}{(1650)} \cos \varphi,$$ or 38 miles; then M is the moon's true place for assumed $\varphi$. (See Note B.) + +Fig. 36. + +A diagram showing a line segment labeled "E" pointing towards "S", with another line segment labeled "a" extending from "E" to "M". Another line segment labeled "b" extends from "a" to "M". A third line segment labeled "c" extends from "b" to "S". The diagram also includes a point labeled "M" and a point labeled "S". + +Thus we see that the moon's P.I. orbit, considered as described about E, and relatively to ES regarded as stationary, is a peculiar epicyclic curve; EC is the radius of the deferent circle turning progressively with its constant angular velocity. It revolves as in Fig. 37, on the line ab, which remains parallel to ES and itself, which also carries at its end the radius bM of the + +138 +THE MOON'S PARALLACTIC INEQUALITY. + +epicycle, which radius rotates progressively with twice the angular velocity of $K_a$. +If we take the step E.F., from E sunward, equal to $a\theta$, then the P.I. orbit will be, relatively to F, a simple epicycle with the same deferent and epicycle having the same simple proportion of their angular velocities. + +The mean motion of the moon in her P.I. orbit can be represented in another manner, which, of course, is the same at bottom, but has its own interest. It follows quite easily from equations (1) and (2). (See Note C.) + +It is often said simply that the moon's P.I. in elongation from the sun is proportional to the sine of her elongation. The difference involved between this and equation (1) is exceedingly small at first, and becomes progressively more inestimable. Let us here take leave to write equation (1) thus + +$$e = \cos^2 - 5\sin e \cdot \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\ e = R(1 + 5\cos e)\cdot R(1 + e\cos e),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\\ e = R(1 + 50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\cdot R(1 + e)\cos e,\qquad(4)$$ + +it becomes a convenient polar equation of the P.I. referred to the (annually rotating) line of conjunction, as prime vector. This equation always, quasi correct, correct, and at syzygies and at quadratures, is true. + +According to this, the moon is at her mean distance from the earth at both quadratures, at her greatest distance at conjunction, and at her least at opposition. These differ from the mean by only about 67-8 miles. + +The P.I. orbit, as given by equation (4), differs very little indeed from a circle which has been first shifted bodily sunwards by the distance R 5332, or 67-8 miles, retaining quite unaltered + +THE MOON'S PARALLACTIC INEQUALITY. + +its syzygy diameter, and then drawn out at right angles to that diameter until it becomes wider by 1c., or only 102 feet. (See Norn D.) The greatest width is very near, and on the sunward side of, the line of quadratures. In the drawing out, the circle becomes flattened a little at both syzygies; but very slightly more at opposition than at conjunction. + +It is easy to obtain geometrically the radius of curvature at conjunction, viz., $\frac{1+e}{1-e}$, and that at opposition, viz., $\frac{1-2e}{1+2e}$; e being the coefficient $e^2$. The former is less than the latter (though by only about 1/4 inch); but both exceed II, the mean radius-vector and radius of curvature. (See Norn E.) + +As with the Var. diagram, so now, we can fill in all the writing in the P.L. diagram, Fig. 35, p. 153, when we know how to draw it. We shall find that the moon's velocity has mean place at first quadrature and most before it at last quadrature; she being of course at her mean place at both syzygies. Among these conclusions let us note particularly that the moon's velocity is least at conjunction, greatest at opposition, and at its mean at both quadratures. We must return to this hereafter. + +The reader will perceive better the differences between the E. and F. diagrams by comparing them for himself than by reading our descriptions alone. We may however, draw his attention to the following point. If we start from C in both diagrams, we shall find that, as regards the writing only, the four reaches, or divisions (constituting one half) of the Var. orbit from C to A, correspond, respectively, to the four reaches (constituting the whole) of the P.L. orbit. + +An elegant explanation of the production of the Parallactic Inequality by the disturbing forces now in question will be found in Airy's *Gravitation*, p. 68, which we shall not reproduce here. (See Norn F.) + +The existence of this lunar inequality was pointed out by Newton. It is very interesting to find that he had determined + +138 +THE MOON'S PARALLACTIC INEQUALITY. + +dynamically its amount with a wonderful closeness of approxi- +mation; though it had not been, in his time, detected by observation. +The value that he gave to the coefficient was 2'00'; this is too large ; the reason being that he went on the supposi- +tion that the sun's parallax was 10', which we now know to be +greater than the true magnitude. +This brings us to the connection between the sun's parallax +and this lunar inequality, which was named by Newton from +its dependence on the ratio between the sun's and the moon's +parallaxes. The coefficient of the P.L. longitude coefficient from the then supposed magnitude of the sun's parallax. But now that the said coefficient is obtainable by observation, it can be used for solving the inverse problem, +viz., obtaining the parallax of the sun. Different formulae have +been given for the connection of the two quantities, which formulae are very nearly, very approximately the same at bottom. +They are so to this, that they give the same ratio of magnitude of the quantities concerned the sun's parallax is almost exactly one fourteenth of the P.L. longitude coefficient. + +But besides this, the interest of this lunar inequality is greatly increased by it having several apparent paradoxes connected with it. This circumstance has attracted great attention it +develops ; and the neglect of it has given rise to certain errone- +ous statements. Of the seeming paradoxes we shall mention five, to be dealt with by eqn. (3) and (4) and Norm E. +1. Since the Var. forces produce the inequalities indicated in Fig. 29, p. 105, the reader might naturally expect that the increase of those forces in the sunward half of the moon's orbit, +the decrease in its earthward half would tend to diminish and +increase the inequalities in the moon's motion there ; and, cor- +respondingly, that the diminution of those forces on the off side +of the orbit, by applying to them the oppositely directed P.I. +forces, should diminish the inequalities there. But these are +both the reverse of the truth. + +A diagram showing a graph with two curves representing Var. forces on opposite sides of a circle. + +THE MOON'S PARALLACTIC INEQUALITY. +139 + +2. Since the Var. orbit is compressed at A and C by the influ- +ence of the Var. forces, the reader would naturally expect that +the just mentioned increase of the forces on the sunward side of +the orbit, by the addition of the P.I. forces, should increase +the compression there ; and, correspondingly, that the diminution of +the forces on the moonward side of the orbit, by the subtraction of +the P.I. forces, should diminish the compression there. But these +are both the reverse of the truth. + +3. When the reader has teachably accepted, from equation (4), +the position that the effect of the P.I. forces is to elongate +the originally undisturbed orbit towards the sun and to compress it +on the opposite side, he will loyally endeavour to carry out his +newly acquired knowledge, and will conclude that the orbit is +now further from the sun on its sunward side than it was stationed +on the side next the sun, than elsewhere. But this is not true; the +sunward side this is the reverse of the truth. (See again Note E.) + +4. The reader will most naturally, and even commendably, +think that the moon would be gaining, or losing, velocity, in the +P.I. orbit, according as the P.I. tangential disturbing forces are +directed towards or away from her. This is true; but there- +fore that her velocity is greatest at conjunction. He will be +confirmed in this expectation by seeing that such happens to be +the case in the Var. orbit; see Fig. 26. He will think also that +the moon's velocity must be least at opposition, since she has +been opposed by the solar tangential forces all the time of her +passing from conjunction to opposition. But all this is the reverse +of the truth. The moon always quickens or slackens her pace in apparent defence of the solar P.I. tangential forces. +(See Note O.) + +5. It would be reasonable enough to expect that since the +P.I. forces are proportional to the inverse fourth power of the +sun's distance, the P.I. longitude should also be proportional +to the same, or at least nearly so; yet it is not so; however it by no means does so. The P.I. longitude is inversely propor- +ted. + +A diagram showing a lunar orbit with points labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. +A diagram showing a lunar orbit with points labeled A', B', C', D', E', F', G', H', I', J', K', L', M', N', O', P', Q', R', S', T', U', V', W', X', Y', Z'. +A diagram showing a lunar orbit with points labeled A'', B'', C'', D'', E'', F'', G'', H'', I'', J'', K'', L'', M'', N'', O'', P'', Q'', R'', S'', T'', U'', V'', W'', X'', Y'', Z''. +A diagram showing a lunar orbit with points labeled A''', B''', C''', D''', E''', F''', G''', H''', I''', J''', K''', L''', M''', N''', O''', P''', Q''', R''', S''', T''', U''', V''', W''', X''', Y''', Z''. +A diagram showing a lunar orbit with points labeled A'''', B'''', C'''', D'''', E'''', F'''', G'''', H'''', I'''', J'''', K'''', L'''', M'''', N'''', O'''', P'''', Q'''', R'''', S'''', T'''', U'''', V'''', W'''', X'''', Y'''', Z''''. +A diagram showing a lunar orbit with points labeled A'''''", B'''''", C'''''", D'''''", E'''''", F'''''", G'''''", H'''''", I'''''", J'''''", K'''''", L'''''", M'''''", N'''''", O'''''", P'''''", Q'''''", R'''''", S'''''", T'''''", U'''''", V'''''", W'''''", X'''''", Y'''''", Z'''''". +A diagram showing a lunar orbit with points labeled A'"""", B'"""", C'"""", D'"""", E'"""", F'"""", G'"""", H'"""", I'"""", J'"""", K'"""", L'"""", M'"""", N'"""", O'"""", P'"""", Q'"""", R'"""", S'"""", T'"""", U'"""", V'"""", W'"""", X'"""", Y'"""", Z'""". +A diagram showing a lunar orbit with points labeled A"""""", B"""""", C"""""", D"""""", E"""""", F"""""", G"""""", H"""""", I"""""”, J"””, K"””, L"””, M"””, N"””, O"””, P"””, Q"””, R"””, S"””, T"””, U"””, V"””, W"””, X"””, Y"””, Z"””. +A diagram showing a lunar orbit with points labeled A'"","B'"","C'"","D'"","E'"","F'"","G'"","H'"","I'"","J'"","K'"","L'"","M'"","N'"","O'"","P'"","Q'"","R'"","S'"","T'"","U'"","V'"","W'"","X'"","Y'"","Z'"." +A diagram showing a lunar orbit with points labeled A"'","B"'","C"'","D"'","E"'","F"'","G"'","H"'","I"'","J"'","K"'","L"'","M"'","N"'","O"'","P"'","Q"'","R"'","S"'","T"'","U"'","V"'","W"'","X"'","Y"'","Z'". +A diagram showing a lunar orbit with points labeled A"','B"',C"',D"',E"',F"',G"',H"',I"',J"',K"',L"',M"',N"',O"',P"',Q"',R"',S"',T"',U"',V"',W"',X"',Y"',Z''. +A diagram showing a lunar orbit with points labeled A"",B"",C"",D"",E"",F"",G"",H"",I"",J"",K"",L"",M"",N"",O"",P"",Q"",R"",S"",T"",U"",V"",W"",X"",Y"",Z"". +A diagram showing a lunar orbit with points labeled A""""",                                                                             .</div> +A diagram showing a lunar orbit with points labeled A""",B",C",D",E",F",G",H",I",J",K",L",M",N",O",P",Q",R",S",T",U",V",W",X",Y",Z".</div> +A diagram showing a lunar orbit with points labeled A""",B""",C""",D""",E""",F""",G""",H""",I""",J""",K""",L""",M""",N""",O""",P""",Q""",R""",S""",T""",U""",V""",W""",X""",Y""",Z"".</div> +A diagram showing a lunar orbit with points labeled A'",B'",C'",D'",E'",F'",G'",H'",I'",J'",K'",L'",M'",N'",O'",P'",Q'",R'",S'",T'",U'",V'",W'",X'",Y'",Z''.</div> +A diagram showing a lunar orbit with points labeled A,"B,"C,"D,"E,"F,"G,"H,"I,"J,"K,"L,"M,"N,"O,"P,"Q,"R,"S,"T,"U,"V,"W,"X,"Y,"Z."</div> +A diagram showing a lunar orbit with points labeled A',"B',"C',"D',"E',"F',"G',"H',"I',"J',"K',"L',"M',"N',"O',"P',"Q',"R',"S',"T',"U',"V',"W',"X',"Y',"Z"."</div> +A diagram showing a lunar orbit with points labeled A",&nb + +140 +THE MOON'S PARALLACTIC INEQUALITY. + +tional to a complicated function of the sun's distance; which function is much nearer to the cube than to the fourth power thereof. So that if there were an alteration of the sun's mean distance, the P.I. and the Var., in longitude would change at not very different rates. +We shall mention further on what some might regard, at first sight, as another apparent paradox; but it is only kinematical in character. +The reader, if he accepts our statements, will probably begin in despair to imagine that the name which has been given to this scheme of lunar inequalities is a mistake for "Paradoxical Inequality." +The general explanation of the above apparent paradoxes is two-fold. In the first place, as we have noted already, the Var., by its disturbance of whatever is the Variance of the P.I., produce their respective inequalities in the moon's motion in two quite different ways, viz., by their direct local influence on the moon's velocity in the various parts of her orbit, and also by what we may call their indirect general influence in deforming the orbit, and thus creating tangential components of the earth's attraction on the moon, which are actually greater than the solar tangential forces. In the second place, while the case of the Variation is simple, because it consists of two equal and opposite forces has but one axis of symmetry passing through the earth, that of the line of syzygies. This involves a most important difference as to the dynamics of these two schemes of lunar inequalities, as considered in Nore F. +It so happens, as we have seen, that, in the case of the Var. orbit, the created terrestrial tangential forces always act along with the solar tangential forces; and thus, in the usual elemental treatment of the Variation, they are not prominently noticed, or are even disregarded altogether; although they are, even in that orbit, more important than are the solar ones themselves, as to their direct local action. + +THE MOON'S PARALLACTIC INEQUALITY. +141 + +But in the case of the P.I. orbit, the relative importance of the terrestrial tangential forces is much more striking, for two reasons. The small solar P.I. forces have, by accumulation of effects, deformed the orbit to such an extent (very small, however, absolutely) that the terrestrial tangential forces created thereby are much greater, proportionally, than the solar tangen- +tial forces. These forces are increased by 30% when multiplied by 30% each. They are, therefore, never less than 3-31 times as great as the latter; and when the moon is not far from quadratures, very much more, proportionally, than this. (See Note II.) And as the greater terrestrial, act always against the smaller solar, ones, the singular result follows that the inequalities in the moon's velocity and in her elongation, now seen only in the P.I. orbit, will be greatly increased by solar tangential forces with which we are now engaged, are liable to occur to effect by their direct local action. So that, paradoxical as it sounds, it is strictly true that the terrestrial tangential forces are the immediate cause of the moon's P.I. in elongation, and that this lunar perturbation would be greater, but for the hindrance of the direct local action of the solar tangential forces. Thus the moon's Parallactic Inequality presents a peculiarly interesting dynamical problem. + +Norr A., from p. 133.—The verification of these expressions for the disturbing forces, though of a simple character, is a little troublesome. If the reader should undertake it, let him beware not to stop at the first approximation, which would give the numerical factor 7 instead of 6; which latter is sufficiently accurate. +SIR, +The value of the coefficient $q_{\text{E}}$ is 0-000858. +These expressions show that the P.I. forces are, quasi propr., inversely proportional to the fourth power of the sun's distance from the earth. It may be just at first sight that they are directly proportional to the first power of this same distance from the earth. But they are proportional only to the + +142 + +THE MOON'S PARALLACTIC INEQUALITY. + +second power thereof. The $M$ comes in on account of the earth's mean attraction on the moon being here taken as unity. + +The trigonometrical factor in the expression for the tangential force can be written $\frac{1}{2}\sin 2\cos e$, that for the radial force can be written $\frac{1}{2}\cos 2\cos e$. This gives the interesting result that, at the elongation $e$, the tangential force divided by the radial force = tan $2e$. + +Note B, from p. 135.—In Fig. 37 the points marked $a$, $b$, $M$, are the same as those similarly marked in Fig. 36. Ea being the moon's mean radius-vector, necessarily drawn vastly too + +Fig. 37. + +short relatively to the other lines, and $a$ the moon's mean place, +draw $ac$ making the angle $\phi$ with the production of $Ea$ (and parallel with ES) to represent $\frac{1}{2}\sin R$; then draw $c$ parallel to $Ea$, and sensibly pointing backwards to the earth; draw $ad$ at right angles to Ea and $cd$; then $ad$ is $\frac{1}{2}\sin R\sin \varphi$. Take $e$ so that $ac$ may represent $\frac{1}{2}\sin R$; draw $CM$ parallel to $ad$; then $dM$ is $\frac{1}{2}R\cos \varphi$, and $M$ is the moon's true place. Now $ec$ is $\frac{1}{2}R\cos \varphi$, and $M$ is the moon's true place. Now bisect it in $b$, and draw $bM$. Then $be$ and $bM$ are both $\frac{1}{2}R(\frac{1}{2}\cos \varphi - \frac{1}{2})$, and $ab$ is $\frac{1}{2}R + \frac{1}{2}R(\frac{1}{2}\cos \varphi - \frac{1}{2})$, or + +THE MOON'S PARALLACTIC INEQUALITY. +143 + +$\frac{1}{2}R(\cos\phi+\sin\phi)$. The angle $Mh=2Mo$, or $2\varphi$; whence the statement in text follows. + +Norm C, from p. 136.-(See Fig. 38.) As before, E is the place of the earth, and S the direction of the sun, and the points marked $a$, $b$, $c$, $d$, represent the moon's mean position in Fig. 37. Let Ez be the moon's mean (both as to position and magnitude) radius-vector; so that $a$ is the moon's mean position. With centre $a$ and radius $\frac{1}{2}R$, describe the circle + +Fig. 38. + +shown in the Fig. Draw the radius parallel to ES, making the angle $hae$ equal to aES, or $\varphi$. Through e draw $fM$ perpendicular to ES; then $af$ is $\frac{1}{2}\sqrt{2}\cos\varphi$, or (since EF is not sensibly different from EE) the change in the length of the moon's radius-vector, for assumed $\varphi$; and $fe$ is $\frac{1}{2}\sqrt{2}\sin\varphi$. Now if $fM$ be to $fe$ in the proportion of the two P.L. coefficients $2'\ 5'$, or, in circular measure, $\frac{\pi}{2}$ to $\frac{\pi}{4}$ which is 15 to 7, very nearly, then M is the position of the moon for her assumed mean elongation $\varphi$, + +$$E \rightarrow S$$ + +144 +THE MOON'S PARALLACTIC INEQUALITY. + +The point M describes, round $a$ as centre, an ellipse which is as though it were rigidly attached to Eo, and therefore rotates about its centre once a month progressively; its semi-axes major and minor, $ay$ and $az$, being $\frac{1}{\sqrt{3}}$R and $\frac{1}{\sqrt{6}}$R, respectively; and as $ae$ rotates, relatively to $ab$, with the moon's mean angular velocity in elongation (not in longitude), the ellipse is described retrogressively in a synodical month. At the time of conjunction the moon is at the point $h$ of the ellipse, and farthest from the earth; at the time of opposition she is at $a$ in the ellipse, and nearest to the earth; and she describes the ellipse with a simple harmonic motion relatively to each of the principal axes of the (rotating) ellipse. + +We have been considering the matter from the standpoint of an observer on the earth. But it is only as seen from the earth that the moon makes a complete circuit round $a$. Since the above ellipse, which is described once only in a synodical month retrogressively, rotates progressively once in the same time, or when observed from any other point on the earth than $A$, the moon never makes any circuit round $a$ relatively to fixed space, or as viewed by a spectator looking at right angles to the plane of her orbit. She is always more or less nearly on the sunward side of $a$. + +Here, then, is the seeming kinematical paradox, as some might regard it at first sight (only), to which we have already alluded; viz., that although the moon's mean place is always nearly on the same side, speaking roughly, of her mean place. The explanation of this is that $a$ is the moon's mean place relatively only to the earth about which she is revolving. + +The P.I. has, moreover, its own seeming kinematical paradox precisely similar to that of the Var. considered at p. 121. + +In this we have contemplated the P.I. as existing by itself; but if we consider it as superposed on the Var., $a$ must be conceived as moving relative to both Var. and P.I. orbit. The inequacy involved in doing this is quite insensible. + +The P.I. disturbing forces are too complicated to be introduced with advantage into Fig. 38. + +The moon's parallactic inequality. 145 + +Norn D, from p. 137.—This will appear thus.—Adopting equation (4), we have $r = R(1 + e \cos \varphi)$, $e$ being the coefficient sine the ordinate at any point of the lunar P.I. orbit is $r \sin \alpha$, or $\left(R(1 + e \cos \varphi)\right) \sin \alpha$. Hence it is $R(\sin \alpha + e \sin 2\alpha)$. Therefore $d y / d x = (c + e - c + e^2) dy/dx$. For $y$ at maximum, $\cos \alpha = -c \cos 2\varphi = -(2 \cos \varphi - 1)$. This quadratic equation gives +$$\cos \alpha = \pm \sqrt{\frac{1}{2} + \frac{1}{16e^2}}$$ +which is +$$\pm \frac{\sqrt{8e^2 + 1} - 4e}{4e},$$ +and this is $c, q_{max}$, or $p_{max}$, +and $\sin \alpha = \sqrt{1 - e^2}$, +which is $1 - j^2$, $q_{max}$, on account of the infinitesimal smallness of $e$. Hence if $r$ sin $\alpha$ is at maximum, in Eq. $1 + e^2)(1 - j^2)$, $\sin \alpha$ is $(R(1 + e^2))^{1/2}$. + +Therefore, taking $R(1 + e^2)$ as 208,800 miles, that maximum diameter is longer than the syzygy diameter, or 2R, by 72°, or 102 feet (Q.E.D.). Said maximum diameter passes very nearly indeed through the middle point of the syzygy diameter, and consequently between the centre of the earth and the sun, and thus does not coincide with the line of quadratures. + +Norn E, from p. 137.—The radius of curvature at conjunction may be obtained thus.—Let $a$ be an indefinitely small $e$ or elongation; then we have by equation (4), for the radius-vector at conjunction, $R(1 + e)$; and being the coefficient $\frac{3}{5}$ as in last Norn. The radius of curvature $p = a/2(fall from tangent).$ +But +$$a^2 = R(1 + e)^2 \sin^2 \alpha,$$ +and +$$2 fall = 2\left(\frac{R(1 + e)}{\cos \alpha} - R(1 + e)\cos \alpha\right),$$ +$$= 2R\left(\frac{1}{\cos \alpha}(1 + e - c - e - c\cos e)\right).$$ + +146 + +THE MOON'S PARALLACTIC INEQUALITY. + +Therefore + +$$\rho = \frac{R(1+c)(1-\cos e)\cos e}{2(1+\cos e + q(1-\cos e))}$$ + +$$= \frac{R(1+c)(1+\cos e)\cos e}{2(1+q(1+\cos e))}$$ + +When $e$ vanishes this becomes $\frac{R(1+c)}{1+2c}$. Similarly, the radius of curvature at opposition $\rho'$ becomes $\frac{R(1-c)}{1-2c}$, as in text. + +These evidently differ very slightly compared from R, and from each other. Taking R as 328,820 miles, and c as 0.000284, we find $\rho' - p = 1.39$ cm. + +Note I, from p. 157—If we give the proof here we should have to do it as it is, which would require much space. But we may make the following observations on the subject. The mode of production of the P.I. orbit is exceedingly different from that of the V. arr. The scheme of Var. forces is symmetrical relatively to the line of syzygies and to that of quadratures; consequently they go through their period of change in half a synodical lunation. But the scheme of P.I. forces is symmetrical relatively to the line of syzygies only, and consequently their period is double that of Var. The focal lines and the scheme of changing of the gravitational forces therein are about equal on one axis only, viz. the apsidal, and the period of the changing forces is that of one revolution of the body about the centre of force. It is evident, therefore, that if there were nothing in the conditions of the case to prevent it, the P.I. forces, when turned into action, would at first originally cause linear orbits, would then go on increasing indefinitely by deformation of the body, whatever the character thereof might be. A moment's consideration will show the nature of the deformation. Since the tangential forces are proportional to $e - \sin e$, the magnitudes of those belonging to the lower two arrows in Fig. 30 vary symmetrically on each side of the quadrature D; they are equal at equal distances + +THE MOON'S PARALLACTIC INEQUALITY. +147 + +on both sides of that point. Therefore, since they are so ex- +ceedingly small, their impulses are very nearly equivalent, for +our present purpose, to a short sufficiently strong tangential +impulse acting at D. Therefore, as they are acting with the +moon's motion, they tend to produce an apogee at the opposite +side of the orbit very near B. Similarly, as the forces belonging +to the two upper arrows in the same Fig. are acting against the +moon's motion, their impulses would produce a perigee very near +D. The outwardly-directed radial forces on the outward side +of the orbit tend to produce an apogee near B, and the inwardly- +directed ones on the other side a perigee near D. The conditions of the focal elliptical orbit lend themselves compliantly to this ; +and if the disturbing forces cease to act, the deformation of the +orbit would continue (with a very slight alteration). The conse- +quence is that if the P.L. forces could continue to act, +until the moon had reached its perigee upon the earth, the +eccentricity of the orbit would go on increasing to a result which +could not easily be followed out ; but probably until the moon +fell upon the earth; the line of apes remainining in quadratures +and fixed in space. But this latter is prevented by the sun's +relative annual revolution round the earth, which would diminish +the eccentricity, and thereby give to the line of apes a pre- + + +A diagram showing three arrows labeled A, B, and C. Arrow A points upwards and to the right, arrow B points downwards and to the left, and arrow C points downwards and to the right. Arrows A and C are connected by a line labeled "de S" pointing downwards and to the left. + + +Fig. 30. + +B + +C + +E + +A + +de S + +D + +148 +THE MOON'S PARALLACTIC INEQUALITY. + +gressive angular movement, relatively to space, which would, at first, be slower than the sun's; so that the sun would be over- +taking it. By the time that the sun had overtaken it, the angular movement of the axis, which had been increasing, though all the while less than that of the sun, would be brought up to equality with that of the sun; and it would therefore continue pointing to the sun. Thus, owing to the buffling conditions of the sun's relative revolution round the earth, the result of the action of the moon on the earth will be a "parallactic inequality" which it would otherwise have. Nevertheless these forces are always tending to produce their own proper effect, which is to make an apogee very near B, and a perigee near D. But the very small effect that they can produce in one lunation, when compounded with that at conjunction, is only sufficient to cause the latter to alter its place by about 30 minutes at preceding lunation, and to keep it moving progressively with the sun. + +The P.L. orbit has been, for convenience, and indeed in accord- +ance with precedent, roughly spoken of as an ellipse; it being intended that the earth is at the focus, and that the apsidal diameter (in syzygy) is the axis-major, with the apogee in conjunction. The velocity of the moon in the P.L. orbit would accord very nearly indeed with this; but the actual "ellipse" is not of such a peculiar kind, in that its axis-major is slightly less than its width. + +Norn G, from p. 130.—This follows from the scheme of P.I. +forces in Fig. 35, p. 133; and from what is told us by equation (3), as mentioned in p. 137, taken in connection with the prin- +ciples of the apparent kinematical synodic index analysis. YLL., p. 121. +But I do not think we feel it difficult to see why this may be well to put the argument together here, though it be a little repetition. By equation (3), the moon is at her mean place at conjunction; therefore she is there moving either with greatest or least velocity—which? At the preceding quadrature D she is, by said equation, more before her mean place, and therefore + +THE MOON'S PARALLACTIC INEQUALITY. + +moving with mean velocity; but, since she is back at her mean place at conjunction, she has been losing velocity in the quadrant preceding conjunction. Therefore she is going with least velocity at conjunction. Similarly, on comparing first quadrature with opposition, we find that the moon is going with greatest velocity at opposition. Thus her velocity is at the maximum at opposition, at the mean at last quadrature, and at the minimum at conjunction; so that she gains velocity all through the semi-orbit; though the solar tangential force has been all the while acting in consequence, or along with her motion. Similarly she gains velocity all through the other semi-orbit; though the solar tangential force has been acting in antecedent, or against her motion. + +We have, so far, been content with the general law of the change of the moon's angular velocity in the P.L. orbit; but the exact law can be obtained by differentiating equation (3). The coefficient, in that equation, as expressed in circular measure, being $\frac{d\varphi}{dt}$ the actual vel., $= \lambda(1 - 2e\cos\varphi)$. This equation shows, at a glance, that the velocity is least at conjunction, greatest at opposition, and at its mean at quadratures; the opposite of what the solar tangential forces would cause by their immediate local action. + +Here $N$, from p. 141—i.e. Fig. 40, E is the earth's place, the curve $c$ a portion of the moon's P.L. orbit, and $de$ an indefinitely small increase of $e$, the moon's elongation. Let $\theta$ be the angle between the moon's radius-vector and the curve at $a$, or $e$, the tangent thereto. + +Then $R$, the moon's mean radius-vector, being taken as unity we have, for the actual radius-vector at $a$, by equation (4), + +$$r = R + e\cos\varphi,$$ + +$$dr = -e\sin\varphi.$$ + +The earth's mean attraction on the moon being, as well as $R$, taken for unity, the earth's attraction at $a$ is + +$$\frac{1}{2} - 2e\cos\varphi.$$ + +150 +THE MOON'S PARALLACTIC INEQUALITY. + +quadrarior, or sensibly 1. This multiplied by $\cos \theta$ is the earth's tangential force at $a$. + +$\cos \theta = \cot \theta$, $q$, $pr$, as $\theta$ differs so very slightly from a right angle. But $\cos \theta = \frac{ab}{bc} = \frac{dr}{de}$ and this is $(1 + c\cos e)de$ or $-c\sin e$, $q$, $pr$, or $-0.000284\sin e$ while the sun's tan- + +Fig. 40. + +F. E. C + +gential force at $a$ is $+0.000585(\sin e - \sin e)$ (see Note A). Therefore the terrestrial is to the solar tangential force, at the same point with the elongation $e$, as 3:31 sec to 1 (Q. E. D.). At quadratures, then, the terrestrial is infinitely greater, proportionally, than the solar tangential force; but this it can easily be, since at those points it is at its maximum, while the solar force is there zero. + +Printed by TAYLOR and FRANCIS, 601 Lion Court, Fleet Street. + +A blank page with a light beige background. + +A blank page with a light beige background. + + + + + + +
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