# Theory and Application of Augmented Dimensional Analysis

Dan Jonsson  Dan Jonsson, University of Gothenburg, Gothenburg, Sweden
<EMAIL_ADDRESS>
###### Abstract.

We present an innovative approach to dimensional analysis, based on a
representation theorem for complete quantity functions with a scaling-
covariant scalar representation. This new theorem, grounded in a purely
algebraic theory of quantity spaces, allows the traditional $\pi$ theorem to
be restated in an explicit and precise form and its prerequisites to be the
clarified and relaxed. Examples of dimensional analysis based on the new
approach are given, showing that results obtained by traditional dimensional
analysis can be strengthened. For example, the orbital period of a two-body
system can be derived without use of equations of motion. The connection
between dimensional analysis and matroid theory is also explored.

##  1\. Introduction

The central theorem in dimensional analysis is the so-called $\pi$ theorem,
with a long history featuring contributions by Fourier [8], Vaschy [20],
Federman [7], Buckingham [4] and others. The $\pi$ theorem shows how to
transform a ”physically meaningful” equation

(1.1) $t_{0}=\phi\mathopen{}\mathclose{{}\left(t_{1},\ldots,t_{n}}\right),$

describing a relationship among quantities, into a more informative equation.
This is done by representing $\phi$ as a product of the form
$\prod_{j=1}^{r}\nolimits\\!x{}_{j}^{k_{j}}\,\psi$ so that, specifically,

(1.2)
$t_{0}=\prod_{j=1}^{r}\nolimits\\!x{}_{j}^{k_{j}}\,\psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n-r}}\right),$

where $\pi_{i}$, defined by

(1.3) $\pi_{i}=y_{i}\prod_{j=1}^{r}\nolimits\\!x{}_{j}^{-k_{ij}},$

is a ”dimensionless product” and
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{r}}\right\\},\mathopen{}\mathclose{{}\left\\{y_{1},\ldots,y_{n-r}}\right\\}}\right\\}$
is a suitable partition of
$\mathopen{}\mathclose{{}\left\\{t_{1},\ldots,t_{n}}\right\\}$.

In traditional dimensional analysis, $\phi$ is a real function, and
$t_{0},\ldots,t_{n}$ are measures of physical quantities. Various assumptions
pertaining to the functions $\phi$, $\psi$, the variables $t_{i}$ and the
parameters $k_{j},k_{ij}$ have been made in connection with the development of
the theory of dimensional analysis:

  1. (1)

Proofs of the $\pi$ theorem rely on assumptions that $\phi$, and hence $\psi,$
have certain mathematical properties. Generally speaking, these assumptions
fall into two groups:

    1. (a)

Those that translate the fact that $\phi$ is “physically meaningful” into
mathematical terms, for example, the assumption proposed by Langhaar [14] and
Brand [2] that $\phi$ satisfies a generalized homogeneity condition.

    2. (b)

Those that are made for internal mathematical reasons. For example, proofs of
the $\pi$ theorem have involved assumptions that $\phi$ is continuous or
differentiable or analytic [9, pp. 67–71]; such assumptions may relate more to
availability of mathematical notions and techniques than to what is
“physically meaningful”.

  2. (2)

It is usually assumed that $t_{i}>0$. This assumption is necessary in some
proofs of the $\pi$ theorem, and avoids the anomaly that terms such as
$\sqrt{-1}$ are not real numbers, but it restricts the scope of dimensional
analysis.

  3. (3)

The exponents $k_{j}$ and $k_{ij}$ are usually assumed to be rational or real
numbers [5, p. 293], but Quade [15] and more recently Raposo [17] use integer
exponents.

  4. (4)

It is often implicitly assumed that for any $\phi$ there is just one $\psi$
such that (1.2) holds, or at least that it suffices to consider one $\psi$ or,
at the very least, deal with one $\psi$ at a time. This way of thinking is
challenged in [10] (see also [11, 21]).

The emergence of quantity calculus [1], and more recent related developments
[6, 15, 23, 5, 10, 16, 12], have led to attempts to formulate versions of the
$\pi$ theorem where $t_{0},\ldots,t_{n}$ are the quantities measured rather
than their numerical measures [6, 15, 23, 19, 17, 13]. The present
contribution belongs to this quantity calculus tradition. Rather than real
functions $\phi$ and $\psi$ we use quantity functions $\Phi$ and $\Psi$ on a
quantity space $Q$ over a field $K$ (see Section 2). In this new context, too,
assumptions of the type highlighted above need to be addressed. It turns out
that it is possible to extend the scope and increase the power of traditional
dimensional analysis.

  1. (i)

$\Phi$ is required to be a quantity function that is algebraically equivalent
to a “complete” quantity function $\Phi^{*}$ with a “scaling-covariant scalar
representation” (see Section 2). This assumption formalizes the idea that
$\Phi$ is a “physically meaningful” quantity function.

  2. (ii)

$Q$ can be a quantity space over any field, so the (generalised) measures of
the quantities $t_{0},\ldots,t_{n}$ need not be real numbers. The quantities
$x_{1},\ldots,x_{r}$ in (1.2) and (1.3) are assumed to be non-zero, but there
are no further absolute constraints on $t_{0},\ldots,t_{n}$ or their measures
(see the discussion in connection with Proposition 1).

  3. (iii)

As only integers are used in exponents, we write $y^{2}=x$ instead of
$y=x^{\frac{1}{2}}$ etc. Specifically, in (1.1) and (1.2) we replace $t_{0}$
by $t_{0}^{\kappa}$, where $\kappa$ is a positive integer, in (1.2) $k_{j}$
are integers, and in (1.3) $k_{ij}$ are integers and $y_{i}$ is replaced by
$y_{i}^{k_{i}}$, where $k_{i}$ are positive integers.

  4. (iv)

There may be more than one partition
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{r}}\right\\},\mathopen{}\mathclose{{}\left\\{y_{1},\ldots,y_{n-r}}\right\\}}\right\\}$
of $\mathopen{}\mathclose{{}\left\\{t_{1},\ldots,t_{n}}\right\\}$ such that
there is a quantity function $\Psi$ and $\kappa$, $k_{j}$, $k_{i}$, $k_{ij}$
such that $t_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits
x{}_{j}^{k_{j}}\,\Psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi{}_{n-r}}\right)$,
where $\pi_{i}=y_{i}^{k_{i}}\prod_{j=1}^{r}\\!x_{j}^{-k_{ij}}$. Thus, the
augmented dimensional analysis developed here does not yield a single equation
as in the conventional approach but a system of $\Theta\geq 1$ equations

$\begin{cases}t_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits
x{}_{1,j}^{k_{1,j}}\,\Psi_{1}\mathopen{}\mathclose{{}\left(\pi_{1,1},\ldots,\pi{}_{1,n-r}}\right),\\\
\cdots\\\ t_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits
x{}_{\Theta,j}^{k_{\Theta,j}}\,\Psi_{\Theta}\mathopen{}\mathclose{{}\left(\pi_{\Theta,1},\ldots,\pi{}_{\Theta,n-r}}\right).\end{cases}$

Section 2, after this Introduction, is devoted to preliminaries. Then the main
representation theorems are stated and proved in Section 3, and applied to
problems of dimensional analysis in Section 4. Several examples of dimensional
analysis are given in Section 5, Section 6 connects dimensional analysis to
matroid theory, and Section 7 explores covariance of scalar representations.

######  Remark 1.

In this article, we mainly describe an “unbalanced” approach to dimensional
analysis, where a dependent variable is chosen at the outset. In Section 6,
however, we will also consider a “balanced” approach, where the choice of
dependent variable is suspended.

##  2\. Preliminaries

###  2.1. Quantity spaces and dimension groups

For the sake of completeness, let us first briefly review some definitions and
results from the theory of quantity spaces [10, 12].

A _scalable monoid_ over a ring $R$ is a monoid $Q$ equipped with an
$R$-action $\cdot$ on $Q$,

$R\times Q\rightarrow
Q,\qquad\mathopen{}\mathclose{{}\left(\alpha,x}\right)\mapsto\alpha\cdot x,$

compatible with multiplication in $Q$. For any $\alpha,\beta\in R$ and $x,y\in
Q$, we have $1\cdot x=x$, $\alpha\cdot\mathopen{}\mathclose{{}\left(\beta\cdot
x}\right)=\alpha\beta\cdot x$, and $\alpha\cdot
xy=\mathopen{}\mathclose{{}\left(\alpha\cdot
x}\right)y=x\mathopen{}\mathclose{{}\left(\alpha\cdot y}\right)$; as a
consequence, $\mathopen{}\mathclose{{}\left(\alpha\cdot
x}\right)\mathopen{}\mathclose{{}\left(\beta\cdot y}\right)=\alpha\beta\cdot
xy$. We denote the identity element of $Q$ by $1_{\\!Q}$, and set
$x^{0}=1_{\\!Q}$ for any $x\in Q$. An element $x\in Q$ may have an inverse
$x^{-1}\in Q$ such that $xx^{-1}=x^{-1}x=1_{\\!Q}$.

A finite _basis_ for a scalable monoid $Q$ is a set
$E_{Q}=\mathopen{}\mathclose{{}\left\\{e_{1},\ldots,e_{m}}\right\\}$ of
invertible elements of $Q$ such that every $x\in Q$ has a unique expansion

$x=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(x}\right)\cdot\prod_{j=1}^{m}\nolimits
e_{j}^{k_{j}},$

where $\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(x}\right)\in R$ and $k_{j}$
are integers. A (finitely generated) _quantity space_ is a commutative
scalable monoid $Q$ over a field $K$, such that there exists a finite basis
for $Q$. The elements of a quantity space are called _quantities_. We may
think of $\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(x}\right)$ as the measure
of $x$ relative to the derived unit $\prod_{j=1}^{m}\nolimits e_{j}^{k_{j}}$
in a coherent system of units, and indirectly relative to the base units in
$E_{Q}$.

The relation $\sim$ on $Q$ defined by $x\sim y$ if and only if $\alpha\cdot
x=\beta\cdot y$ for some $\alpha,\beta\in K$ is a congruence on $Q$. The
corresponding equivalence classes are called _dimensions_ ;
$\mathopen{}\mathclose{{}\left[x}\right]$ is the dimension that contains $x$.
We have $\mathopen{}\mathclose{{}\left[\lambda\cdot
x}\right]=\mathopen{}\mathclose{{}\left[x}\right]$ for any $\lambda\in K,x\in
Q$. The set of all dimensions in $Q$, denoted $Q/{\sim}$, is a finitely
generated free abelian group with multiplication defined by
$\mathopen{}\mathclose{{}\left[x}\right]\mathopen{}\mathclose{{}\left[y}\right]=\mathopen{}\mathclose{{}\left[xy}\right]$
and identity $\mathopen{}\mathclose{{}\left[1_{Q}}\right]$. Hence,
$\mathopen{}\mathclose{{}\left[x^{\kappa}}\right]=\mathopen{}\mathclose{{}\left[x}\right]^{\kappa}$
for any integer $\kappa.$

The quantities in a dimension $\mathsf{C}\in Q/{\sim}$ form a one-dimensional
vector space over $K$ with addition inherited from $K$ and a unique zero
vector or _zero_ _quantity_ $0_{\mathsf{C}}\neq 1_{Q}$. While
$0_{\mathsf{C}}x=0_{\mathsf{C}\mathopen{}\mathclose{{}\left[x}\right]}$ for
every $x\in Q$, the product of non-zero quantities is a non-zero quantity. A
quantity is invertible if and only if it is non-zero, and any $u\in\mathsf{C}$
such that $u\neq 0_{\mathsf{C}}$ is a _unit quantity_ for $\mathsf{C}$,
meaning that for every $x\in\mathsf{C}$ there is a unique $\mu\in K$ for $u$
such that $x=\mu\cdot u$, where $\mu=0$ if and only if $x=0_{\mathsf{C}}$.

###  2.2. Quantity functions, prebases and local bases

A (dimensional)_quantity function_ on a quantity space $Q$ over a field $K$ is
a function of the form

(2.1)
$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{n}\rightarrow\mathsf{C}_{0},\qquad\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)\mapsto
q_{0}\qquad(\mathsf{C}_{i}\in Q/{\sim}).$

Equation (1.1) now takes the form
$q_{0}=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$, where
$q_{0},\ldots,q_{n}$ are quantities.

The dimensions $\mathsf{D}_{1},\ldots,\mathsf{D}_{\nu}\in Q/{\sim}$ or
quantities $x_{1},\ldots,x_{\nu}\in Q$ are said to be _dependent_ when there
are integers $\epsilon_{1},\ldots,\epsilon_{\nu}$, not all equal to zero, such
that
$\mathsf{D}_{1}^{\epsilon_{1}}\cdots\mathsf{D}_{\nu}^{\epsilon_{\nu}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
or $\mathopen{}\mathclose{{}\left[x_{1}^{\epsilon_{1}}\cdots
x_{\nu}^{\epsilon_{\nu}}}\right]=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$,
respectively. Dimensions or quantities which are not dependent are said to be
_independent_.

A ___prebasis of dimensions_ for $\Phi$ is a subset
$\mathsf{E}=\mathopen{}\mathclose{{}\left\\{\mathsf{E}_{1},\ldots,\mathsf{E}_{r}}\right\\}$
of
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{n}}\right\\}$
which is maximal among the subsets of
$\mathsf{F}=\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{0},\mathsf{C}_{1},\ldots,\mathsf{C}_{n}}\right\\}$
with independent dimensions, meaning that for every
$\mathsf{C}{}_{i}\in\mathsf{F}$ there are integers
$\kappa_{i},\kappa_{i1},\ldots,\kappa_{ir}$ such that $\kappa_{i}\neq 0$ and

(2.2)
$\mathsf{C}_{i}^{\kappa_{i}}=\prod_{j=1}^{r}\nolimits\mathsf{E}_{j}^{\kappa_{ij}}.$

A quantity function for which there is at least one prebasis of dimensions is
said to be _precomplete_.

Similarly, a ___prebasis_ (of quantities) for $\Phi$ is a set of non-zero
quantities $E=\mathopen{}\mathclose{{}\left\\{e_{1},\ldots,e_{r}}\right\\}$
such that
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[e_{1}}\right],\ldots,\mathopen{}\mathclose{{}\left[e_{r}}\right]}\right\\}$
is a prebasis of dimensions for $\Phi$, meaning that
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[e_{1}}\right],\ldots,\mathopen{}\mathclose{{}\left[e_{r}}\right]}\right\\}\subseteq\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{n}}\right\\}$
and for every $q_{i}\in\mathsf{C}_{i}\in\mathsf{F}$ there are integers
$\kappa_{i}^{\prime},\kappa_{i1}^{\prime},\ldots,\kappa_{ir}^{\prime}$ such
that $\kappa_{i}^{\prime}\neq 0$ and

(2.3)
$\mathopen{}\mathclose{{}\left[q_{i}^{\kappa_{i}^{\prime}}}\right]=\prod_{j=1}^{r}\nolimits\mathopen{}\mathclose{{}\left[e_{j}}\right]^{\kappa_{ij}^{\prime}}=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}^{\prime}}}\right].$

Let
$\mathsf{E}=\mathopen{}\mathclose{{}\left\\{\mathsf{E}_{1},\ldots,\mathsf{E}_{r}}\right\\}$
be a prebasis of dimensions for $\Phi$ and
$E=\mathopen{}\mathclose{{}\left\\{e_{1},\ldots,e_{r}}\right\\}$ a set of
quantities. If $0_{\mathsf{E}_{j}}\neq e_{j}\in\mathsf{E}_{j}$ for
$j=1,\ldots.r$ (henceforth $E\precsim\mathsf{E}$), then $E$ is a prebasis for
$\Phi$ and (2.2) implies (2.3), with $\kappa_{i}=\kappa_{i}^{\prime}$ and
$\kappa_{ij}=\kappa_{ij}^{\prime}$, since
$\mathsf{C}_{i}^{\kappa_{i}}=\prod_{j=1}^{r}\nolimits\mathsf{E}_{j}^{\kappa_{ij}}$
implies
$\mathopen{}\mathclose{{}\left[q_{i}^{\kappa_{i}}}\right]=\mathopen{}\mathclose{{}\left[q_{i}}\right]^{\kappa_{i}}=\prod_{j=1}^{r}\nolimits\mathopen{}\mathclose{{}\left[e_{j}}\right]^{\kappa_{ij}}=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}}}\right]$.

It can be shown [12] that in a set containing all sequences of exponents
$\kappa_{i}^{*},\kappa_{i1}^{*},\ldots,\kappa_{ir}^{*}$ satisfying (2.2) or
(2.3), there is a unique sequence of _canonical exponent_ s
$\overline{\kappa}_{i}^{*},\overline{\kappa}_{i1}^{*},\ldots,\overline{\kappa}_{ir}^{*}$
such that $\overline{\kappa}_{i}^{*}>0$ and
$\mathrm{gcd}\mathopen{}\mathclose{{}\left(\overline{\kappa}_{i}^{*},\overline{\kappa}_{i1}^{*},\ldots,\overline{\kappa}_{ir}^{*}}\right)=1$.

If $\overline{\kappa}_{i}^{*}=1$ for any sequence of canonical exponents given
by a prebasis of dimensions $\mathsf{E}$ or a prebasis $E$ then $\mathsf{E}$
is said to be a _local basis of dimensions_ and $E$ is said to be a _local
basis_ (of quantities). A quantity function with a local basis of dimensions
is said to be _complete_. It follows from the corresponding result for
prebases that if $\mathsf{E}$ is a local basis of dimensions for $\Phi$ and
$E\precsim\mathsf{E}$, then $E$ is a local basis for $\Phi$ and (2.2) implies
(2.3), with $\overline{\kappa}_{i}=\overline{\kappa}_{i}^{\prime}=1$ and
$\overline{\kappa}_{ij}=\overline{\kappa}_{ij}^{\prime}$.

It is shown in Appendix A that every $q_{i}\in\mathsf{C}_{i}\in\mathsf{F}$ has
a unique expansion
$q_{i}=\mu_{E}\mathopen{}\mathclose{{}\left(q_{i}}\right)\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}}$ relative to every local basis $E$ for $\Phi$, so a local
basis is indeed a basis $E_{Q^{\prime}}$ for a certain subspace $Q^{\prime}$
of $Q$. Furthermore, for any $E$,
$\mu_{E}\mathopen{}\mathclose{{}\left(qq^{\prime}}\right)=\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)\mu_{E}\mathopen{}\mathclose{{}\left(q^{\prime}}\right)$
for any $q,q^{\prime}$, and
$\mu_{E}\mathopen{}\mathclose{{}\left(q^{-1}}\right)=\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)^{-1}$
for any invertible $q$ [12]. Also, if
$q\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ then the unique expansion of
$q$ is $q=\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)\cdot 1_{Q}$ for any
$E$ since $\prod_{j=1}^{r}e_{j}^{0}=1_{Q}$ for any $e_{j}$, so
$\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)$ does not depend on $E$.

###  2.3. Covariant scalar representations

The “physically meaningful” quantity functions of interest in augmented
dimensional analysis have scalar representations that do not depend on a
choice of local basis. Specifically, a _covariant scalar representation_ of a
complete quantity function $\Phi$ of the form (2.1) is a function
$\phi:K^{n}\rightarrow K$ such that

(2.4)
$\mu_{E}\mathopen{}\mathclose{{}\left(\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)}\right)=\phi\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)$

for any local basis $E$ for $\Phi$ and any $q_{1},\ldots,q_{n}$.

If $E$ is only required to be any local basis such that $E\precsim\mathsf{E}$,
where $\mathsf{E}$ is a fixed local basis of dimensions, then $\phi$ is said
to be a _scaling-covariant_ scalar representation. A covariant scalar
representation is obviously scaling-covariant.

For example, the identity map $\mathrm{id}:K\rightarrow K$ is a covariant
scalar representation of the identity map
$\mathrm{Id}:\mathsf{C}\rightarrow\mathsf{C}$ since
$\mu_{E}\mathopen{}\mathclose{{}\left(\mathrm{Id}\mathopen{}\mathclose{{}\left(q}\right)}\right)=\mathrm{id}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)}\right)$
for all $q$ and $E$. Also, if $\phi_{1}$ is a covariant scalar representation
of $\Phi_{1}:\mathsf{C}_{1}\rightarrow\mathsf{D}_{1}$, and $\phi_{2}$ of
$\Phi_{1}:\mathsf{C_{2}}\rightarrow\mathsf{D}_{2}$, then $\phi_{1}\phi_{2}$ is
a covariant scalar representation of $\Phi_{1}\Phi_{2}$ since

$\mu_{E}\mathopen{}\mathclose{{}\left(\Phi_{1}\mathopen{}\mathclose{{}\left(q_{1}}\right)\Phi_{2}\mathopen{}\mathclose{{}\left(q_{2}}\right)}\right)=\mu_{E}\mathopen{}\mathclose{{}\left(\Phi_{1}\mathopen{}\mathclose{{}\left(q_{1}}\right)}\right)\mu_{E}\mathopen{}\mathclose{{}\left(\Phi_{2}\mathopen{}\mathclose{{}\left(q_{2}}\right)}\right)=\phi_{1}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{1}}\right)}\right)\phi_{2}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{2}}\right)}\right)$

for all $q_{1},q_{2}$ and $E$, and if $\phi$ is a covariant scalar
representation of
$\Phi:\mathsf{D}_{1}\times\cdots\times\mathsf{D}_{n}\rightarrow\mathsf{D}_{0}$
and $\omega_{1},\ldots,\omega_{n}$ are covariant scalar representations of
$\Omega_{1},\ldots,\Omega_{n}$, where
$\text{$\Omega$}_{i}:$$\mathsf{C}_{i}\rightarrow\mathsf{D}_{i}$, then
$\phi\circ\mathopen{}\mathclose{{}\left(\omega_{1},\ldots,\omega_{n}}\right)$
is a scalar representation of
$\Phi\circ\mathopen{}\mathclose{{}\left(\Omega_{1},\ldots,\Omega_{n}}\right)$
since

$\mu_{E}\mathopen{}\mathclose{{}\left(\Phi\mathopen{}\mathclose{{}\left(\Omega_{1}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\Omega_{n}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)}\right)=\phi\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(\Omega_{1}\mathopen{}\mathclose{{}\left(q_{1}}\right)}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(\Omega_{n}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)}\right)=\phi\mathopen{}\mathclose{{}\left(\omega_{1}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{1}}\right)}\right),\ldots,\omega_{n}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)}\right)$

for all $q_{1},\ldots,q_{n}$ and $E$. These results apply to scaling-covariant
scalar representations as well.

Note that a complete quantity function need not have a covariant scalar
representation.

######  Example 1.

Every quantity function
$\Phi:\mathsf{C}\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right]$, where
$\mathsf{C}\neq\mathopen{}\mathclose{{}\left[1_{Q}}\right]$, has a local basis
$\mathopen{}\mathclose{{}\left\\{e}\right\\}$, where $0_{\mathsf{C}}$ $\neq
e\in\mathsf{C}$, since $\mathsf{C}=\mathsf{C}^{1}$ and
$\mathopen{}\mathclose{{}\left[1_{Q}}\right]=\mathsf{\mathsf{C}^{0}}$. If we
set $\Phi_{u}\mathopen{}\mathclose{{}\left(\lambda\cdot u}\right)=\lambda\cdot
1_{Q}$ for some $0_{\mathsf{C}}\neq u\in\mathsf{C}$ then
$\Phi_{u}\mathopen{}\mathclose{{}\left(u}\right)=1\cdot 1_{Q}$ and
$\Phi_{u}\mathopen{}\mathclose{{}\left(2\cdot u}\right)=2\cdot 1_{Q}$ but
$u=1\cdot u$ and $2\cdot u=1\cdot\mathopen{}\mathclose{{}\left(2\cdot
u}\right)$. Hence,
$\mu_{\mathopen{}\mathclose{{}\left\\{u}\right\\}}\mathopen{}\mathclose{{}\left(\Phi_{u}\mathopen{}\mathclose{{}\left(u}\right)}\right)\neq\mu_{\mathopen{}\mathclose{{}\left\\{2\cdot
u}\right\\}}\mathopen{}\mathclose{{}\left(\Phi_{u}\mathopen{}\mathclose{{}\left(2\cdot
u}\right)}\right)$ and
$\phi\mathopen{}\mathclose{{}\left(\mu_{\mathopen{}\mathclose{{}\left\\{u}\right\\}}\mathopen{}\mathclose{{}\left(u}\right)}\right)=\phi\mathopen{}\mathclose{{}\left(\mu_{\mathopen{}\mathclose{{}\left\\{2\cdot
u}\right\\}}\mathopen{}\mathclose{{}\left(2\cdot u}\right)}\right)$ for any
$\phi:K\rightarrow K$, but then
$\mu_{\mathopen{}\mathclose{{}\left\\{u}\right\\}}\mathopen{}\mathclose{{}\left(\Phi_{u}\mathopen{}\mathclose{{}\left(u}\right)}\right)\neq\phi\mathopen{}\mathclose{{}\left(\mu_{\mathopen{}\mathclose{{}\left\\{u}\right\\}}\mathopen{}\mathclose{{}\left(u}\right)}\right)$
or $\mu_{\mathopen{}\mathclose{{}\left\\{2\cdot
u}\right\\}}\mathopen{}\mathclose{{}\left(\Phi_{u}\mathopen{}\mathclose{{}\left(2\cdot
u}\right)}\right)\neq\phi\mathopen{}\mathclose{{}\left(\mu_{\mathopen{}\mathclose{{}\left\\{2\cdot
u}\right\\}}\mathopen{}\mathclose{{}\left(2\cdot u}\right)}\right)$.
Conversely, if
$\Phi:\mathsf{C}\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ has a
covariant scalar representation then $\Phi$ is a constant function, as shown
in Example 4.

##  3\. Representation theorems

Given a quantity function
$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{n}\rightarrow\mathsf{C}_{0}$
and a permutation $\sigma$ of
$\mathopen{}\mathclose{{}\left(1,\ldots,n}\right)$ one can construct a new
quantity function
$\Phi^{\prime}:\mathsf{C}_{\sigma\mathopen{}\mathclose{{}\left(1}\right)}\times\cdots\times\mathsf{C}_{\sigma\mathopen{}\mathclose{{}\left(n}\right)}\rightarrow\mathsf{C}_{0}$
by setting
$\Phi^{\prime}\mathopen{}\mathclose{{}\left(q_{\sigma\mathopen{}\mathclose{{}\left(1}\right)},\ldots,q_{\sigma\mathopen{}\mathclose{{}\left(n}\right)}}\right)=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$.
One may in particular reorder the arguments so that the elements of a certain
prebasis of dimensions for $\Phi$ come first in
$\mathopen{}\mathclose{{}\left(q_{\sigma\mathopen{}\mathclose{{}\left(1}\right)},\ldots,q_{\sigma\mathopen{}\mathclose{{}\left(n}\right)}}\right)$;
we call this a _prebasis reordering_ of
$\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$. The following fact
applies to a quantity function obtained, if necessary, by a prebasis
reordering of arguments.

######  Proposition 1.

_Let $Q$ be a quantity space and $\Phi$ a precomplete quantity function on $Q$
such that_

$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{r}\times\mathsf{D}_{1}\times\cdots\times\mathsf{D}_{n-r}\rightarrow\mathsf{D_{0}^{\kappa}},\qquad\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)\mapsto
y_{0}^{\kappa}\qquad\mathopen{}\mathclose{{}\left(n,r\geq 0}\right),$

_where $y_{0}\in\mathsf{D}_{0}$ and $\kappa$ is a positive integer, and let
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{r}}\right\\}$
be a prebasis of dimensions for $\Phi$ with corresponding canonical exponents
$k_{i},k_{i1},\ldots,k_{ir}$, so that $\mathsf{D}_{0}^{\kappa
k_{0}}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{0j}}$ and
$\mathsf{D}_{i}^{k_{i}}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{ij}}$ for
$i=1,\ldots,n-r$. If $k_{0}=1$ and there exists a bijection of arguments_

$\chi:\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)\mapsto\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)\qquad\mathopen{}\mathclose{{}\left(x_{i}\in\mathsf{C}_{i},y_{i}\in\mathsf{D}_{i}}\right),$

_then there exists a complete quantity function_

$\Phi^{*}:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{r}\times\mathsf{D}_{1}^{k_{1}}\times\cdots\times\mathsf{D}_{n-r}^{k_{n-r}}\rightarrow\mathsf{D}_{0}^{\kappa},\qquad\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)\mapsto
y_{0}^{\kappa},$

_with a local basis of dimensions
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{r}}\right\\}$,
and such that_

(3.1)
$\Phi^{*}\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)=\Phi\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right).$

###### Proof.

Note that if $y_{i}\in\mathsf{D}_{i}$ then
$y_{i}^{k_{i}}\in\mathsf{D}_{i}^{k_{i}}$ since
$\mathopen{}\mathclose{{}\left[q^{k_{i}}}\right]=\mathopen{}\mathclose{{}\left[q}\right]^{k_{i}}$.
If we set $\Phi^{*}=\Phi\circ\chi^{-1}$ then (3.1) holds, and
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{r}}\right\\}$
is a local basis of dimensions for $\Phi^{*}$ since
$\mathsf{D}_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits\\!\mathsf{C}_{j}^{k_{0j}}$
and
_$\mathsf{D}_{i}^{k_{i}}=\prod_{j=1}^{r}\nolimits\\!\mathsf{C}_{j}^{k_{ij}}$
_for _ $i=1,\ldots,n-r$, _with corresponding canonical exponents
_$1,k_{i1},\ldots,k_{ir}$_ for $i=0,\ldots,n-r$.__ ∎

A precomplete quantity function $\Phi$ with $k_{0}=1$ and a bijective $\chi$
is said to be _solvable_. Each of the following three conditions guarantees
the existence of the bijection $\chi$:

  1. (1)

$n=r$.

  2. (2)

$k_{i}=1$ for $i=1,\ldots,n-r$.

  3. (3)

The restriction of $\chi$ to $\mathsf{D}_{i}$ is bijective for
$i=1,\ldots,n-r$; this generalizes (2).

In proofs of scalar versions of the $\pi$ theorem it is often assumed that
$\mu_{E}\mathopen{}\mathclose{{}\left(q_{i}}\right)>0$ for all $q_{i}$, so
that we can let
$\chi^{-1}\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(q_{i}}\right)^{\kappa}}\right)$
be the positive $\kappa$th root of
$\mu_{E}\mathopen{}\mathclose{{}\left(q_{i}}\right)^{\kappa}$. Versions of the
$\pi$ theorem for an ordered quantity space could be qualified by similarly
assuming that $q_{i}>0_{\mathopen{}\mathclose{{}\left[q_{i}}\right]}$ for all
$q_{i}$. Proposition 1 suggests, however, that a positive sign condition is
somewhat redundant and beside the point. (We may use a quantity space over
$\mathbb{C}$ to eliminate another reason for excluding roots of negative real
numbers.)

Instead of restricting the domains of $y_{1},\ldots,y_{n-r}$, we may define
bijections by “splitting” $\Phi^{*}$ into well-defined functions
$\Phi_{i}^{*}$. For example, if
$\Phi\mathopen{}\mathclose{{}\left(x,y}\right)\neq\Phi\mathopen{}\mathclose{{}\left(x,-y}\right)$
for $y\neq 0_{\mathopen{}\mathclose{{}\left[y}\right]}$ whereas
$\Phi^{*}\mathopen{}\mathclose{{}\left(x,y^{2}}\right)=\Phi^{*}\mathopen{}\mathclose{{}\left(x,\mathopen{}\mathclose{{}\left(-y}\right)^{2}}\right)$
for all $y$ then we cannot define $\Phi^{*}$ by setting
$\Phi^{*}=\Phi\circ\chi^{-1}$, but

$\displaystyle\chi_{1}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x,y^{2}}\right)\qquad(y>0_{\mathopen{}\mathclose{{}\left[y}\right]}),$
$\displaystyle\chi_{2}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x,y^{2}}\right)\qquad(y<0_{\mathopen{}\mathclose{{}\left[y}\right]})$

are bijections, and we can set $\Phi_{1}^{*}=\Phi\circ\chi_{1}^{-1}$ and
$\Phi_{2}^{*}=\Phi\circ\chi_{2}^{-1}$, so that
$\Phi_{1}^{*}\mathopen{}\mathclose{{}\left(x,y^{2}}\right)=\Phi\mathopen{}\mathclose{{}\left(x,y}\right)$
for $y>0_{\mathopen{}\mathclose{{}\left[y}\right]}$ and
$\Phi_{2}^{*}\mathopen{}\mathclose{{}\left(x,y^{2}}\right)=\Phi\mathopen{}\mathclose{{}\left(x,y}\right)$
for $y<0_{\mathopen{}\mathclose{{}\left[y}\right]}$. Out of $\Phi_{1}^{*}$ and
$\Phi_{2}^{*}$, only one function remains if we require that
$y>0_{\mathopen{}\mathclose{{}\left[y}\right]}$ but also if we require that
$y<0_{\mathopen{}\mathclose{{}\left[y}\right]}$.

######  Theorem 1 (special $\pi$ theorem).

Let $Q$ be a quantity space over $K$, and let

(3.2)
$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{r}\times\mathsf{D}_{1}\times\cdots\times\mathsf{D}_{n-r}\rightarrow\mathsf{D}_{0},\qquad\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)\mapsto
y_{0}\qquad\mathopen{}\mathclose{{}\left(n,r\geq 0}\right)$

be a complete quantity function on $Q$ such that
$\mathopen{}\mathclose{{}\left\\{\mathsf{C_{1}},\ldots,\mathsf{C}_{r}}\right\\}$
is a local basis of dimensions with corresponding canonical exponents
$1,k_{i1},\ldots,k_{ir}$ so that
$\mathsf{D}_{i}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{ij}}$ for $i=0,\ldots,n-r$.

Then $\Phi$ has a local basis, and if $\Phi$ has a scaling-covariant scalar
representation, then there exists a quantity function of $n-r$ arguments

$\Psi:\mathopen{}\mathclose{{}\left[1_{Q}}\right]\times\cdots\times\mathopen{}\mathclose{{}\left[1_{Q}}\right]\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right]$

such that if all $x_{j}$ are non-zero and
$\pi_{i}=y_{i}^{k_{i}}\prod_{j=1}^{r}\nolimits x_{j}^{-k_{ij}}$ for
$i=0,\ldots,n-r$ then

(3.3)
$\pi_{0}=\Psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n-r}}\right),$

or equivalently

(3.4)
$y_{0}=\prod_{j=1}^{r}\nolimits\\!x_{j}^{k_{0j}}\,\Psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n-r}}\right).$

###### Proof.

Any $E=\mathopen{}\mathclose{{}\left\\{e_{1},\ldots,e_{r}}\right\\}$ such that
$E\precsim\mathsf{C}=\mathopen{}\mathclose{{}\left\\{\mathsf{C_{1}},\ldots,\mathsf{C}_{r}}\right\\}$
is a local basis for $\Phi$, and we have

$y_{i}=\mu_{E}\mathopen{}\mathclose{{}\left(y_{i}}\right)\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{k_{ij}}\qquad\mathopen{}\mathclose{{}\left(i=0,\ldots,n-r}\right)$

as $\mathsf{D}_{i}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{ij}}$ implies
$\mathopen{}\mathclose{{}\left[y_{i}}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}\nolimits
e_{j}^{k_{ij}}}\right]$. Fix
$X=\mathopen{}\mathclose{{}\left\\{x_{1},\ldots,x_{r}}\right\\}$ such that
$X\precsim\mathsf{C}$, and define $\breve{y}_{i}$ by

$\breve{y}{}_{i}=\prod_{j=1}^{r}\nolimits
x_{j}^{k_{ij}}\qquad\mathopen{}\mathclose{{}\left(i=0,\ldots,n-r}\right).$

Recall that all $x_{j}$ are non-zero, so that each $\breve{y}_{i}$ is
invertible and $\breve{y}{}_{i}^{-1}=\prod_{j=1}^{r}\nolimits
x_{j}^{-k_{ij}}$. Hence,
$y_{i}\breve{y}_{i}^{-1}=\mu_{E}\mathopen{}\mathclose{{}\left(y_{i}}\right)\cdot\prod_{j=1}^{r}\nolimits\mathopen{}\mathclose{{}\left(e_{j}x_{j}^{-1}}\right)^{k_{ij}}$,
where
$\mathopen{}\mathclose{{}\left[e_{j}x_{j}^{-1}}\right]=\mathopen{}\mathclose{{}\left[e_{j}}\right]\mathopen{}\mathclose{{}\left[x_{j}}\right]^{-1}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$,
so $y_{i}\breve{y}_{i}^{-1}\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$.
Thus, $\mu_{E}\mathopen{}\mathclose{{}\left(y_{i}\breve{y}_{i}^{-1}}\right)$
does not depend on $E$. Note that
$\mu_{E}\mathopen{}\mathclose{{}\left(\breve{y}_{i}}\right)=\prod_{j=1}^{r}\mu_{E}\mathopen{}\mathclose{{}\left(x_{j}}\right)^{k_{ij}}$
for $i=0,\ldots,n-r$.

Let $\mathbf{q}$ denote the sequence of quantities
$x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}$, and let
$\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)$ be the
sequence of scalars
$\mu_{E}\mathopen{}\mathclose{{}\left(x_{1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(x_{r}}\right),\mu_{E}\mathopen{}\mathclose{{}\left(y_{1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(y_{n-r}}\right)$.
By definition, $y_{0}=\Phi\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)$
and by assumption there is a function $\phi:K^{n}\rightarrow K$ such that
$\mu_{E}\mathopen{}\mathclose{{}\left(\Phi\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right)=\phi\mathopen{}\mathclose{{}\left(\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right)$
for any $\mathbf{q}$ and $E\precsim\mathsf{C}$, so as
$\mu_{E}\mathopen{}\mathclose{{}\left(x_{j}}\right)\neq 0$ for $j=1,\ldots r$
there is a function $\varphi:K^{n}\mapsto K$ such that for any $\mathbf{q}$
and $E\precsim\mathsf{C}$ we have

$\displaystyle\mu_{E}\mathopen{}\mathclose{{}\left(y_{0}\breve{y}_{0}^{-1}}\right)=\frac{\mu_{E}\mathopen{}\mathclose{{}\left(\Phi\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right)}{\mu_{E}\mathopen{}\mathclose{{}\left(\breve{y}_{0}}\right)}=\frac{\phi\mathopen{}\mathclose{{}\left(\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right)}{\prod_{j=1}^{r}\mu_{E}\mathopen{}\mathclose{{}\left(x_{j}}\right)^{k_{0j}}}=\varphi\mathopen{}\mathclose{{}\left(\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right).$

Furthermore, there is, for given $k_{ij}$, a bijection between scalar
sequences

$\displaystyle\omega:\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)\>\longmapsto\>\mu_{E}\mathopen{}\mathclose{{}\left(x_{1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(x_{r}}\right),\frac{\mu_{E}\mathopen{}\mathclose{{}\left(y_{1}}\right)}{\prod_{j=1}^{r}\mu_{E}\mathopen{}\mathclose{{}\left(x_{j}}\right)^{k_{1j}}},\ldots,\frac{\mu_{E}\mathopen{}\mathclose{{}\left(y_{n-r}}\right)}{\prod_{j=1}^{r}\mu_{E}\mathopen{}\mathclose{{}\left(x_{j}}\right)^{k_{\mathopen{}\mathclose{{}\left(n-r}\right)j}}}\>,$

so there is a function $\Gamma=\varphi\circ\omega^{-1}:K^{n}\rightarrow K$
such that

$\mu_{E}\mathopen{}\mathclose{{}\left(y_{0}\breve{y}_{0}^{-1}}\right)=\varphi\mathopen{}\mathclose{{}\left(\boldsymbol{\mu}_{E}\mathopen{}\mathclose{{}\left(\mathbf{q}}\right)}\right)=\Gamma\mathopen{}\mathclose{{}\left(\mu_{E}\mathopen{}\mathclose{{}\left(x_{1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(x_{r}}\right),\mu_{E}\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right),\ldots,\mu_{E}\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{-1}}\right)}\right).$

Note that we can set $E=X$ since $X\precsim\mathsf{C}$. There is thus a
function $\psi:K^{n-r}\rightarrow K$ such that

(3.5)
$\displaystyle\mu_{X}\mathopen{}\mathclose{{}\left(y_{0}\breve{y}_{0}^{-1}}\right)=\Gamma\mathopen{}\mathclose{{}\left(1,\ldots,1,\mu_{X}\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right),\ldots,\mu_{X}\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{n-r}}\right)}\right)=\psi\mathopen{}\mathclose{{}\left(\mu_{X}\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right),\ldots,\mu_{X}\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{n-r}}\right)}\right)$

since $x_{j}=1\cdot x_{j}$ so that
$\mu_{X}\mathopen{}\mathclose{{}\left(x_{j}}\right)=1$ for $j=1,\ldots,r$.

To complete the proof, we convert the scalar function $\psi$, obtained from
the quantity function $\Phi$, back into a quantity function $\Psi$. As
$\mu_{E}\mathopen{}\mathclose{{}\left(y_{i}\breve{y}_{i}^{-1}}\right)$ does
not depend on $E\precsim\mathsf{C}$, we can define a quantity function of
$n-r$ arguments

$\Psi:\mathopen{}\mathclose{{}\left[1_{Q}}\right]\times\cdots\times\mathopen{}\mathclose{{}\left[1_{Q}}\right]\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right],$

which depends only on $\psi$, by setting

$\Psi\mathopen{}\mathclose{{}\left(\mu_{X}\\!\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right)\cdot
1_{Q},\ldots,\mu_{X}\\!\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{-1}}\right)\cdot
1_{Q}}\right)=\psi\mathopen{}\mathclose{{}\left(\mu_{X}\\!\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right),\ldots,\mu_{X}\\!\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{-1}}\right)}\right)\cdot
1_{Q},$

so that, by (3.5),

(3.6)
$\mu_{X}\mathopen{}\mathclose{{}\left(y_{0}\breve{y}_{0}^{-1}}\right)\cdot
1_{Q}=\Psi\mathopen{}\mathclose{{}\left(\mu_{X}\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1}}\right)\cdot
1_{Q},\ldots,\mu_{X}\mathopen{}\mathclose{{}\left(y_{n-r}\breve{y}_{n-r}^{-1}}\right)\cdot
1_{Q}}\right).$

Recall that
$y_{i}\breve{y}_{i}^{-1}=\mu_{X}\mathopen{}\mathclose{{}\left(y_{i}\breve{y}_{i}^{-1}}\right)\cdot
1_{Q}$ is the unique expansion of $y_{i}\breve{y}_{i}^{-1}$ relative to $X$
since $y_{i}\breve{y}_{i}^{-1}\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$,
so we can rewrite (3.6) as

(3.7)
$y_{0}\breve{y}_{0}^{-1}=\Psi\mathopen{}\mathclose{{}\left(y_{1}\breve{y}_{1}^{-1},\ldots,y_{n-r}\breve{y}_{n-r}^{-1}}\right).$

In view of the fact that $\pi_{i}=y_{i}\prod_{j=1}^{r}\nolimits
x_{j}^{-k_{ij}}$, we have $y_{i}\breve{y}_{i}^{-1}=\pi_{i}$, so as required we
have proved (3.3), or equivalently (3.4). ∎

If
$\Phi\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)=\breve{y}_{0}\Psi\mathopen{}\mathclose{{}\left(\pi{}_{1},\ldots,\pi{}_{n-r}}\right)=\breve{y}_{0}\Psi^{\prime}\mathopen{}\mathclose{{}\left(\pi{}_{1},\ldots,\pi{}_{n-r}}\right)$,
where $\breve{y}_{0}=\prod_{j=1}^{r}\nolimits x_{j}^{k_{0j}}$, and
$x_{1},\ldots,x_{r}$ are non-zero then
$\Psi\mathopen{}\mathclose{{}\left(\pi{}_{1},\ldots,\pi{}_{n-r}}\right)=\Psi^{\prime}\mathopen{}\mathclose{{}\left(\pi{}_{1},\ldots,\pi{}_{n-r}}\right)$
since $\breve{y}_{0}$ is invertible, so the representation in terms of $\Psi$
of $\Phi$ restricted to non-zero values of $x_{1},\ldots,x_{r}$ is unique.

We can now combine Proposition 1 and Theorem 1 into a general representation
theorem.

######  Theorem 2 (general $\pi$ theorem).

_Let $Q$ be a quantity space and $\Phi$ a precomplete quantity function on $Q$
such that_

$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{r}\times\mathsf{D}_{1}\times\cdots\times\mathsf{D}_{n-r}\rightarrow\mathsf{D_{0}^{\kappa}},\qquad\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)\mapsto
y_{0}^{\kappa}\qquad\mathopen{}\mathclose{{}\left(n,r\geq 0}\right),$

_where $y_{0}\in\mathsf{D}_{0}$ and $\kappa$ is a positive integer, and let
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{r}}\right\\}$
be a prebasis of dimensions for $\Phi$ with corresponding canonical exponents
$k_{i},k_{i1},\ldots,k_{ir}$, so that $\mathsf{D}_{0}^{\kappa
k_{0}}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{0j}}$ and
$\mathsf{D}_{0}^{k_{i}}=\prod_{j=1}^{r}\mathsf{C}_{j}^{k_{ij}}$ for
$i=1,\ldots,n-r$. If $\Phi$ is solvable then there then there exists a
complete quantity function_

$\Phi^{*}:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{r}\times\mathsf{D}_{1}^{k_{1}}\times\cdots\times\mathsf{D}_{n-r}^{k_{n-r}}\rightarrow\mathsf{D}_{0}^{\kappa},\qquad\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)\mapsto
y_{0}^{\kappa},$

_with a local basis of dimensions
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{1},\ldots,\mathsf{C}_{r}}\right\\}$,
and such that_

(3.8)
$\Phi^{*}\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)=\Phi\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right),$

_and if $\Phi^{*}$ _has a scaling-covariant scalar representation, then there
exists a quantity function of $n-r$ arguments

$\Psi:\mathopen{}\mathclose{{}\left[1_{Q}}\right]\times\cdots\times\mathopen{}\mathclose{{}\left[1_{Q}}\right]\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right]$

such that if all $x_{j}$ are non-zero and
$\pi_{i}=y_{i}^{k_{i}}\prod_{j=1}^{r}\nolimits x_{j}^{-k_{ij}}$ for
$i=0,\ldots,n-r$, then

(3.9)
$\displaystyle\pi_{0}=\Psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n-r}}\right),$

or equivalently

(3.10)
$y_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits\\!x_{j}^{k_{0j}}\,\Psi\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n-r}}\right).$

###### Proof.

Apply Proposition 1 to $\Phi$ and then apply Theorem 1 to $\Phi^{*}$, keeping
in mind that
$y_{0}^{k_{0}}=\Phi^{*}\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1}^{k_{1}},\ldots,y_{n-r}^{k_{n-r}}}\right)=\Phi\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{r},y_{1},\ldots,y_{n-r}}\right)$.
∎

##  4\. Doing dimensional analysis

Let $Q$ be a quantity space and $\Phi$ a quantity function on $Q$ such that

$\Phi:\mathsf{C}_{1}\times\cdots\times\mathsf{C}_{n}\rightarrow\mathsf{C_{0}^{\kappa}},\qquad\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)\mapsto
q_{0}^{\kappa},$

where $q_{0}\in\mathsf{C}_{0}$ and $\kappa$ is a positive integer.__ Consider
a corresponding equation

(4.1)
$q_{0}^{\kappa}=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right).$

The purpose of augmented dimensional analysis is to represent (4.1) by a
system of equations of the form (3.10). This can be done, provided that
specific conditions hold, as described below.

Let $\mathscr{G}$ be the subgroup of $Q/{\sim}$ generated by
$\mathopen{}\mathclose{{}\left\\{\mathsf{C}_{0},\mathsf{C}_{1},\ldots,\mathsf{C}_{n}}\right\\}$.
Recall that $Q/{\sim}$ is a free abelian group, so $\mathscr{G}$ is also a
free abelian group. Thus, any two maximal sets of independent elements in
$\mathscr{G}$ has the same number of elements, by definition equal to the rank
$r$ of $\mathscr{G}$. This implies that any two prebases of dimensions for
$\Phi$ has $r$ elements.

If $\Phi$ is precomplete then there is one or more prebasis reorderings
$\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1},\ldots.y_{\theta,n-r}}\right)$
of $\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$ such that
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[x_{\theta,1}}\right],\ldots,\mathopen{}\mathclose{{}\left[x_{\theta,r}}\right]}\right\\}$
is a prebasis of dimensions for $\Phi$ and
$E_{\theta}=\mathopen{}\mathclose{{}\left\\{x_{\theta,1},\ldots,x_{\theta,r}}\right\\}$
a prebasis of non-zero quantities for $\Phi$. Thus, we obtain a system of
$\Theta$ equations

(4.2)
$\begin{cases}y_{0}^{\kappa}=\Phi_{\theta}^{\prime}\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1},\ldots,y_{\theta,n-r}}\right)&\mathopen{}\mathclose{{}\left(\theta=1,\ldots,\Theta}\right),\end{cases}$

where $y_{0}=q_{0}$ and
$\Phi_{\theta}^{\prime}\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1},\ldots,y_{\theta,n-r}}\right)=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$
for all $\theta$.

For each $E_{\theta}$ we have canonical exponents
$k_{0},k_{\theta,01},\ldots,k_{\theta,0r}$ and
$k_{\theta,i},k_{\theta,i1},\ldots,k_{\theta,ir}$ such that

(4.3) $\mathopen{}\mathclose{{}\left[y_{0}}\right]^{\kappa
k_{0}}=\prod_{j=1}^{r}\nolimits\mathopen{}\mathclose{{}\left[x_{\theta,j}}\right]^{k_{\theta,0j}},\qquad\mathopen{}\mathclose{{}\left[y_{\theta,i}}\right]^{k_{\theta,i}}=\prod_{j=1}^{r}\nolimits\mathopen{}\mathclose{{}\left[x_{\theta,j}}\right]^{k_{\theta,ij}}\quad\mathopen{}\mathclose{{}\left(i=1,\ldots,n-r}\right).$

If each $\Phi_{\theta}$ is solvable, so that $k_{0}=1$ and there is a
bijection

$\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1},\ldots,y_{\theta,n-r}}\right)\mapsto\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1}^{k_{\theta,1}},\ldots,y_{\theta,n-r}^{k_{\theta,n-r}}}\right),$

then there is a system of equations with complete functions
$\Phi_{1}^{*},\ldots,\Phi_{\Theta}^{*}$,

(4.4)
$\begin{cases}y_{0}^{\kappa}=\Phi_{\theta}^{*}\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1}^{k_{\theta,1}},\ldots,y_{\theta,n-r}^{k_{\theta,n-r}}}\right)&\mathopen{}\mathclose{{}\left(\theta=1,\ldots,\Theta}\right),\end{cases}$

where
$\Phi_{\theta}^{*}\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1}^{k_{\theta,1}},\ldots,y_{\theta,n-r}^{k_{\theta,n-r}}}\right)=\Phi_{\theta}^{\prime}\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r},y_{\theta,1},\ldots,y_{\theta,n-r}}\right)$
for all $\theta$.

Assume that each $\Phi_{\theta}^{*}$ has a scaling-covariant scalar
representation, and set
$\pi_{\theta,i}=y_{\theta,i}^{k_{\theta,i}}\prod_{j=1}^{r}x_{\theta,j}^{-k_{\theta,ij}}$
for $\theta=1,\ldots,\Theta$ and $i=1,\ldots.n-r$. We then obtain the system
of equations

(4.5) $\begin{cases}y_{0}^{\kappa}=\prod_{j=1}^{r}\nolimits
x_{\theta,j}^{k_{\theta,0j}}\,\Psi_{\theta}\mathopen{}\mathclose{{}\left(\pi_{\theta,1},\ldots,\pi_{\theta,n-r}}\right)&\mathopen{}\mathclose{{}\left(\theta=1,\ldots,\Theta}\right),\end{cases}$

where $\prod_{j=1}^{r}\nolimits
x_{\theta,j}^{k_{\theta,0j}}\,\Psi_{\theta}\mathopen{}\mathclose{{}\left(\pi_{\theta,1},\ldots,\pi_{\theta,n-r}}\right)=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$
for all $\theta$.

Note that we can derive (4.5) from (4.1) by the method described here only if
$\Phi$ is precomplete, all $\Phi_{\theta}^{\prime}$ are solvable, and all
$\Phi_{\theta}^{*}$ have scaling-covariant scalar representations. Otherwise,
we can obtain an incomplete result if $\Phi$ is precomplete and there is some
$\theta$ such that $\Phi_{\theta}$ is solvable and $\Phi_{\theta}^{*}$ has a
scaling-covariant scalar representation.

It remains to explain how to specify the structure of $\mathscr{G}\subseteq
Q/{\sim}$ so that for each $\theta$ we obtain the canonical exponents
$k_{0},k_{\theta,0j}$ and $k_{\theta,i},k_{\theta,ij}$ required in (4.3). In
some elementary cases, this can be done directly; see Examples 3 and 4 in
Section 5. In most cases, however, the specification of the structure of
$\mathscr{G}$ involves a choice of basis for $Q/{\sim}$ as described below.

Given a dimensional problem specified by $Q$ and
$\Phi:\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)\mapsto
q_{0}^{\kappa}$, fix a basis
$\mathsf{E}=\mathopen{}\mathclose{{}\left\\{\mathsf{E}_{1},\ldots,\mathsf{E}_{m}}\right\\}$
for $Q/{\sim}$. Each $\mathopen{}\mathclose{{}\left[q_{i}}\right]$ is
associated with a unique column vector
$\delta_{i}=\mathopen{}\mathclose{{}\left(\delta_{i1},\ldots,\delta_{im}}\right)^{\\!\mathrm{T}}$
of integers $\delta_{i\ell}$ such that
$\mathopen{}\mathclose{{}\left[q_{i}}\right]=\prod_{\ell=1}^{m}\mathsf{E}_{\ell}^{\delta_{i\ell}}$.
The _dimensional matrix_ for $Q$ and $\Phi$ is

(4.6)
$\begin{array}[]{ccccccc}&\mathopen{}\mathclose{{}\left[q_{0}}\right]&\mathopen{}\mathclose{{}\left[q_{1}}\right]&\cdots&\mathopen{}\mathclose{{}\left[q_{i}}\right]&\cdots&\mathopen{}\mathclose{{}\left[q_{n}}\right]\\\
\mathsf{E}_{1}&\delta_{01}&\delta_{11}&\cdots&\delta_{i1}&\cdots&\delta_{n1}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots\\\
\mathsf{E}_{m}&\delta_{0m}&\delta_{1m}&\cdots&\delta_{im}&\cdots&\delta_{nm}\end{array}.$

Consider $\overline{q}_{j}$ where
$\mathopen{}\mathclose{{}\left[\overline{q}_{j}}\right]\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[q_{0}}\right],\ldots,\mathopen{}\mathclose{{}\left[q_{n}}\right]}\right\\}$,
and let
$\overline{\delta}_{j}=\mathopen{}\mathclose{{}\left(\overline{\delta}_{j1},\ldots,\overline{\delta}_{jm}}\right)^{\\!\mathrm{T}}$
be the column vector associated with $\overline{q}_{j}$. It is clear from the
construction of the dimensional matrix that each $\overline{q}_{j}$ is
represented by $\overline{\delta}_{j}$ relative to $\mathsf{E}$ in such a way
that for any $\overline{q}_{1}$,$\ldots,\overline{q}_{\nu}$ we have
$\prod_{j=1}^{\nu}\mathopen{}\mathclose{{}\left[\overline{q}_{j}}\right]^{\kappa_{j}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
if and only if
$\sum_{j=1}^{\nu}\kappa_{j}\overline{\delta}_{j}=\boldsymbol{0}^{m}$, where
$\boldsymbol{0}^{m}$ is a column vector with $m$ zeros – that is, if and only
if $\kappa_{1},\ldots,\kappa_{\nu}$ satisfy the system of equations

(4.7)
$\begin{cases}\sum_{j=1}^{\nu}\nolimits\overline{\delta}_{j\ell}\kappa_{j}=0&\mathopen{}\mathclose{{}\left(\ell=1,\ldots,m}\right).\end{cases}$

Hence, the rank of the dimensional matrix is equal to the rank $r$ of the free
abelian group $\mathscr{G}$.

Considering a function
$\Phi_{\theta}^{\prime}:\mathopen{}\mathclose{{}\left(x_{\theta,1},\ldots,x_{\theta,r}y_{\theta,1},\ldots,y_{\theta,n-r}}\right)\mapsto
y_{0}^{\kappa}$ obtained by a prebasis reordering of the arguments of $\Phi$,
it also follows from the equivalence of
$\prod_{j=1}^{\nu}\mathopen{}\mathclose{{}\left[\overline{q}_{j}}\right]^{\kappa_{j}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
and equations (4.7) that the canonical exponents
$k_{0},k_{\theta,01},\ldots,k_{\theta,0r}$ and
$k_{\theta,i},k_{\theta,i1},\ldots,k_{\theta,ir}$ in (4.3) are the unique
solutions such that $k_{0},k_{\theta,i}>0$ and
$\mathrm{gcd}\mathopen{}\mathclose{{}\left(k_{0},k_{\theta,01},\ldots,k_{\theta,0r}}\right)=\mathrm{gcd}\mathopen{}\mathclose{{}\left(k_{\theta,i},k_{\theta,i1},\ldots,k_{\theta,ir}}\right)=1$
to the $1+n-r$ systems of equations

$\begin{cases}\mathopen{}\mathclose{{}\left\\{\sum_{j=1}^{r}\nolimits\overline{\delta}_{j\ell}k_{\theta,0j}=\overline{\delta}_{0\ell}\kappa
k_{0}}\right.&\mathopen{}\mathclose{{}\left(\ell=1,\ldots,m}\right),\\\
\mathopen{}\mathclose{{}\left\\{\sum_{j=1}^{r}\nolimits\overline{\delta}_{j\ell}k_{\theta,ij}=\overline{\delta}_{i\ell}k_{\theta,i}}\right.&\mathopen{}\mathclose{{}\left(\ell=1,\ldots,m}\right),\qquad\mathopen{}\mathclose{{}\left(i=1,\ldots,n-r}\right),\end{cases}$

where $\overline{\delta}_{01},\ldots,\overline{\delta}_{0m}$,
$\overline{\delta}_{i1},\ldots,\overline{\delta}_{im}$
$\mathopen{}\mathclose{{}\left(i>0}\right)$ and
$\overline{\delta}_{j1},\ldots,\overline{\delta}_{jm}$ correspond to $y_{0}$,
$y_{\theta,i}$ and $x_{\theta,j}$, respectively.

######  Example 2.

Consider the equation
$q_{0}^{2}=\Phi\mathopen{}\mathclose{{}\left(q_{1},q_{q}.q_{3}}\right)$ and
the dimensional matrices

$\begin{array}[]{ccccc}\mathopen{}\mathclose{{}\left(A}\right)&\mathopen{}\mathclose{{}\left[q_{0}}\right]&\mathopen{}\mathclose{{}\left[q_{1}}\right]&\mathopen{}\mathclose{{}\left[q_{2}}\right]&\mathopen{}\mathclose{{}\left[q_{3}}\right]\\\
\mathsf{E}_{1}&2&1&2&1\\\
\mathsf{E}_{2}&1&0&0&1\end{array},\begin{array}[]{ccccc}\mathopen{}\mathclose{{}\left(B}\right)&\mathopen{}\mathclose{{}\left[y_{0}}\right]&\mathopen{}\mathclose{{}\left[x_{1,1}}\right]&\mathopen{}\mathclose{{}\left[x_{1,2}}\right]&\mathopen{}\mathclose{{}\left[y_{1,1}}\right]\\\
\mathsf{E}_{1}&2&1&1&2\\\
\mathsf{E}_{2}&1&0&1&0\end{array},\begin{array}[]{ccccc}\mathopen{}\mathclose{{}\left(C}\right)&\mathopen{}\mathclose{{}\left[y_{0}}\right]&\mathopen{}\mathclose{{}\left[x_{2,1}}\right]&\mathopen{}\mathclose{{}\left[x_{2,2}}\right]&\mathopen{}\mathclose{{}\left[y_{2,1}}\right]\\\
\mathsf{E}_{1}&2&2&1&1\\\ \mathsf{E}_{2}&1&0&1&0\end{array},$

where $\mathopen{}\mathclose{{}\left(B}\right)$ and
$\mathopen{}\mathclose{{}\left(C}\right)$ are obtained from
$\mathopen{}\mathclose{{}\left(A}\right)$ by the prebasis reorderings
$\mathopen{}\mathclose{{}\left(q_{1},q_{2},q_{3}}\right)\mapsto\mathopen{}\mathclose{{}\left(q_{1},q_{3},q_{2}}\right)=\mathopen{}\mathclose{{}\left(x_{1,1},x_{1,2},y_{1,1}}\right)$
and
$\mathopen{}\mathclose{{}\left(q_{1},q_{2},q_{3}}\right)\mapsto\mathopen{}\mathclose{{}\left(q_{2},q_{3},q_{1}}\right)=\mathopen{}\mathclose{{}\left(x_{2,1},x_{2,2},y_{2,1}}\right)$,
respectively. From $\mathopen{}\mathclose{{}\left(B}\right)$ and
$\mathopen{}\mathclose{{}\left(C}\right)$, we obtain

$\binom{4}{2}=2\binom{1}{0}+2\binom{1}{1},\quad\binom{2}{0}=2\binom{1}{0}+0\binom{1}{1},\qquad\binom{4}{2}=\binom{2}{0}+2\binom{1}{1},\quad
2\binom{1}{0}=\binom{2}{0}+0\binom{1}{1},$

so the canonical exponents in (4.3) are

$\displaystyle\mathopen{}\mathclose{{}\left(k_{0},k_{1,01},k_{1,02}}\right)=\mathopen{}\mathclose{{}\left(1,2,2}\right),\qquad\mathopen{}\mathclose{{}\left(k_{1,1},k_{1,11},k_{1,12}}\right)=\mathopen{}\mathclose{{}\left(1,2,0}\right),$
$\displaystyle\mathopen{}\mathclose{{}\left(k_{0},k_{2,01},k_{2,02}}\right)=\mathopen{}\mathclose{{}\left(1,1,2}\right),\qquad\mathopen{}\mathclose{{}\left(k_{2,1},k_{2,11},k_{2,12}}\right)=\mathopen{}\mathclose{{}\left(2,1,0}\right).$

When $k_{0}=1$, the integers $k_{\theta,0j}$ must satisfy
$\sum_{j=1}^{r}\nolimits\overline{\delta}_{j\ell}k_{\theta,0j}=\overline{\delta}_{0\ell}\kappa$
for some $\kappa\geq 1$ and $\ell=1,\ldots,m$. There is a unique minimal
$\kappa$ such that this is possible for $\theta=1,\ldots,\Theta$: the
_canonical_ $\kappa$, equal to 2 in Example 2. Also note that the set of
prebases of dimensions for
$\Phi:\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)\mapsto
q_{0}^{\kappa}$ is an invariant of the dimensional matrix (4.6); it does not
depend on $\kappa$ since
$\mathopen{}\mathclose{{}\left[q_{0}^{\kappa}}\right]^{k_{0}}=\mathopen{}\mathclose{{}\left[q_{0}}\right]^{\mathopen{}\mathclose{{}\left(\kappa
k_{0}}\right)}=\mathopen{}\mathclose{{}\left[q_{0}}\right]^{k_{0}^{\prime}}$
for any $\kappa$.

######  Remark 2.

It is known from the practice of dimensional analysis that a change of the
basis $\mathsf{E}$ for a dimensional matrix may lead to a change of the
canonical exponents associated with its variables. This can only happen,
however, if the change of basis is tied to a change of quantity space. If $Q$
is a fixed quantity space then a change of basis for $Q/{\sim}$ does not
affect the canonical coefficients since the choice of a basis for $Q/{\sim}$
obviously does not affect the structure of $Q/{\sim}$.

##  5\. Examples of dimensional analysis

We start with two theoretically interesting examples of dimensional analysis
without dimensional matrices, and then move on to a little more familiar
applications with dimensional matrices as inputs. In this section, notation
such as
$\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{r}}\right)\mathopen{}\mathclose{{}\left(q_{r+1},\ldots,q_{n}}\right)$
or
$\Phi^{*}\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{r}}\right)\mathopen{}\mathclose{{}\left(q_{r+1}^{k_{1}},\ldots,q_{n}^{k_{n-r}}}\right)$
will be used to set the $r$ elements of a local basis apart from the remaining
arguments, as the renaming of quantities used earlier for this purpose is
impractical in concrete cases. The quantities in the set
$\mathopen{}\mathclose{{}\left\\{q_{1},\ldots,q_{n}}\right\\}$ are understood
to be non-zero, and in general the members of any set regarded as a prebasis
or a local basis are understood to be non-zero quantities. Only quantity
spaces over $\mathbb{R}$ will be considered below.

######  Example 3.

By convention, $\prod_{j=1}^{0}e_{j}^{\kappa_{j}}=1_{Q}$ so if
$q\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ then
$\mathopen{}\mathclose{{}\left[q}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{0}e_{j}^{\kappa_{j}}}\right]$,
so $\emptyset$ is (vacuously) the unique prebasis as well as the unique local
basis for any quantity function $\Phi$ of the form

$\Phi:\mathopen{}\mathclose{{}\left[1_{Q}}\right]\times\cdots\times\mathopen{}\mathclose{{}\left[1_{Q}}\right]\rightarrow\mathopen{}\mathclose{{}\left[1_{Q}}\right],\qquad\Phi:\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{k}}\right)\mapsto
q_{0}.$

Thus, we have
$q_{0}=\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)=\Phi^{\prime}\mathopen{}\mathclose{{}\left(}\right)\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)$
and

$q_{0}^{1}=\Phi^{*}\mathopen{}\mathclose{{}\left(}\right)\mathopen{}\mathclose{{}\left(q_{1}^{1},\ldots,q_{n}^{1}}\right),$

so $\Phi^{*}=\Phi$. Also, $\phi^{*}$ defined by
$\phi^{*}\mathopen{}\mathclose{{}\left(\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)\cdot
1_{Q}=\Phi^{*}\mathopen{}\mathclose{{}\left(}\right)\mathopen{}\mathclose{{}\left(q_{1}^{1},\ldots,q_{n}^{1}}\right)$
is a covariant scalar representation of $\Phi^{*}$ since
$\mu_{\emptyset}\mathopen{}\mathclose{{}\left(\Phi^{*}\mathopen{}\mathclose{{}\left(}\right)\mathopen{}\mathclose{{}\left(q_{1}^{1},\ldots,q_{n}^{1}}\right)}\right)=\mu_{\emptyset}\mathopen{}\mathclose{{}\left(\phi^{*}\mathopen{}\mathclose{{}\left(\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)\cdot
1_{Q}}\right)=\phi^{*}\mathopen{}\mathclose{{}\left(\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)$,
as we have the expansion
$\phi^{*}\mathopen{}\mathclose{{}\left(\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)\cdot
1_{Q}=\mu_{\emptyset}\mathopen{}\mathclose{{}\left(\phi^{*}\mathopen{}\mathclose{{}\left(\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{1}}\right),\ldots,\mu_{\emptyset}\mathopen{}\mathclose{{}\left(q_{n}}\right)}\right)\cdot
1_{Q}}\right)\cdot 1_{Q}$. Hence,

$q_{0}=1_{Q}\Psi\mathopen{}\mathclose{{}\left(q_{1}/1_{Q},\ldots,q_{n}/1_{Q}}\right)=\Psi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right),$

so $\Psi=\Phi$. We conclude that it is possible but pointless to use
dimensional analysis if all quantity variables range over
$\mathopen{}\mathclose{{}\left[1_{Q}}\right]$.

######  Example 4.

Consider a quantity function $\Phi$ of the form

(5.1)
$\Phi:\mathsf{C}_{1}\rightarrow\mathsf{C}_{0},\qquad\Phi\mathopen{}\mathclose{{}\left(q_{1}}\right)\mapsto
q_{0},$

where
$\mathopen{}\mathclose{{}\left[q_{0}}\right]=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
and
$\mathopen{}\mathclose{{}\left[q_{1}}\right]\neq\mathopen{}\mathclose{{}\left[1_{Q}}\right]$,
As
$\mathopen{}\mathclose{{}\left[q_{0}}\right]=\mathopen{}\mathclose{{}\left[q_{1}}\right]^{0}$,
$\mathopen{}\mathclose{{}\left\\{q_{1}}\right\\}$ is a prebasis and local
basis for $\Phi$. We have
$q_{0}=\Phi\mathopen{}\mathclose{{}\left(q_{1}}\right)=\Phi^{\prime}\mathopen{}\mathclose{{}\left(q_{1}}\right)\mathopen{}\mathclose{{}\left(}\right)$
and

$q_{0}=\Phi^{*}\mathopen{}\mathclose{{}\left(q_{1}}\right)\mathopen{}\mathclose{{}\left(}\right),$

so $\Phi=\Phi^{*}$. If $\Phi$ has a scaling-covariant scalar representation
then

$q_{0}^{1}=1_{Q}\Psi\mathopen{}\mathclose{{}\left(}\right)=k\qquad\mathopen{}\mathclose{{}\left(k\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]}\right)$

since $q_{1}^{0}=1_{Q}$, so $\Phi$ is a constant function. Had $\Phi$ not
admitted a scaling-covariant scalar representation then nothing could have
been said about how $\Phi\mathopen{}\mathclose{{}\left(q_{1}}\right)$ depends
on $q_{1}$, so we obtain specific information about $\Phi$ from this
assumption. A non-constant function of the form (5.1), complete but without a
scaling-covariant scalar representation, was defined in Example 1.

In the examples below, every quantity function $\Phi$ is precomplete, and we
assume that every $\Phi_{i}^{*}$ has a scaling-covariant scalar
representation, meaning that every $\Phi_{i}^{*}$ is “physically meaningful”.

######  Example 5.

Consider a (right, circular) cone. The slant height $H$ of the cone (that is,
the length of the line segment from the periphery of its base to its apex) is
determined by the area of its base $a$ and its height $h$,

$H^{\kappa}=\Phi\mathopen{}\mathclose{{}\left(a,h}\right).$

The dimensional matrix for $\Phi$ is

$\begin{array}[]{cccc}&H&a&h\\\ \mathsf{L}&1&2&1\end{array}.$

$\Phi$ has prebases $\mathopen{}\mathclose{{}\left\\{a}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{h}\right\\}$, so the equation system of the
form (4.2) is

$\begin{cases}H^{\kappa}=\Phi_{1}^{\prime}\mathopen{}\mathclose{{}\left(a}\right)\mathopen{}\mathclose{{}\left(h}\right),&\mathopen{}\mathclose{{}\left(C1}\right)\\\
H^{\kappa}=\Phi_{2}^{\prime}\mathopen{}\mathclose{{}\left(h}\right)\mathopen{}\mathclose{{}\left(a}\right).&\mathopen{}\mathclose{{}\left(C2}\right)\end{cases}$

If $\kappa=1$ then $\mathopen{}\mathclose{{}\left(C2}\right)$ is solvable yet
$\mathopen{}\mathclose{{}\left(C1}\right)$ is not: there is no integer
$\alpha$ such that
$\mathopen{}\mathclose{{}\left[H}\right]^{1}=\mathopen{}\mathclose{{}\left[a}\right]^{\alpha}$.
But $\mathopen{}\mathclose{{}\left(C2}\right)$ gives
$H^{1}=\Phi_{2}^{*}\mathopen{}\mathclose{{}\left(h}\right)\mathopen{}\mathclose{{}\left(a^{1}}\right)$,
so

$H=h\Psi_{2}\mathopen{}\mathclose{{}\left(a/h^{2}}\right).$

Here, $\Psi_{2}\mathopen{}\mathclose{{}\left(x}\right)=1_{Q}+\frac{x}{\pi}$
since $h\sqrt{1_{Q}+\frac{\pi r^{2}}{\pi
h^{2}}}=\sqrt{h^{2}\mathopen{}\mathclose{{}\left(1_{Q}+\frac{r^{2}}{h^{2}}}\right)}=\sqrt{h^{2}+r^{2}}=H$
by Pythagoras’ theorem.

If $\kappa=2$ and $h>0_{\mathopen{}\mathclose{{}\left[h}\right]}$ then both
$\mathopen{}\mathclose{{}\left(C1}\right)$ and
$\mathopen{}\mathclose{{}\left(C2}\right)$ are solvable. We have

$\begin{cases}H^{2}=\Phi_{21}^{*}\mathopen{}\mathclose{{}\left(a}\right)\mathopen{}\mathclose{{}\left(h^{2}}\right),\\\
H^{2}=\Phi_{22}^{*}\mathopen{}\mathclose{{}\left(h}\right)\mathopen{}\mathclose{{}\left(a^{1}}\right),\end{cases}$

so

$\begin{cases}H^{2}=a\Psi_{21}\mathopen{}\mathclose{{}\left(h^{2}/a}\right),\\\
H^{2}=h^{2}\Psi_{22}\mathopen{}\mathclose{{}\left(a/h^{2}}\right).\end{cases}$

It is easy to verify that
$\Psi_{21}\mathopen{}\mathclose{{}\left(x}\right)=\frac{1_{Q}}{\pi}+x$ and
$\Psi_{22}=\Psi_{2}^{2}$.

######  Remark 3.

Note that if $q\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ then we may
define $\sqrt{q}$ as
$\sqrt{\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)}\cdot 1_{Q}$, $e^{q}$ as
$e^{\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)}\cdot 1_{Q}$ and so on,
since $\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)$ does not depend on $E$.
Conversely, if $q\notin\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ then
$\sqrt[n]{q}$ has no meaning unless $q=x^{n}$ for some $x\in Q$ and some
positive integer $n$ since in a monoid $x^{n}$ is just a shorthand for
repeated multiplication by $x$. Thus, the present restriction of exponents to
integer values is a necessary consequence of letting dimensional analysis
apply to quantities rather than scalars.

######  Example 6.

Assume that the period of oscillation $t$ of a pendulum depends on its length
$\ell$, the mass of the bob $m$, the amplitude of the oscillation $\theta$ (an
angle) and the constant of gravity $g$, that is,

$t^{\kappa}=\Phi(\ell,m,\theta,g),$

and let the dependencies among the corresponding dimensions be given by the
dimensional matrix

$\begin{array}[]{cccccc}&\mathopen{}\mathclose{{}\left[t}\right]&\mathopen{}\mathclose{{}\left[\ell}\right]&\mathopen{}\mathclose{{}\left[m}\right]&\mathopen{}\mathclose{{}\left[\theta}\right]&\mathopen{}\mathclose{{}\left[g}\right]\\\
\mathsf{L}&0&1&0&0&1\\\ \mathsf{T}&1&0&0&0&-2\\\
\mathsf{M}&0&0&1&0&0\end{array}.$

The unique prebasis for $\Phi$ is
$\mathopen{}\mathclose{{}\left\\{\ell,m,g}\right\\}$. If $\kappa=2$ then
$\Phi$ is solvable, and the prebasis for $\Phi$ is a local basis for
$\Phi^{*}$ with

$t^{2}=\Phi^{*}\mathopen{}\mathclose{{}\left(\ell,m,g}\right)\mathopen{}\mathclose{{}\left(\theta^{1}}\right),$

since
$\mathopen{}\mathclose{{}\left[t}\right]^{2}=\mathopen{}\mathclose{{}\left[\ell}\right]^{1}\mathopen{}\mathclose{{}\left[m}\right]^{0}\mathopen{}\mathclose{{}\left[g}\right]^{-1}$
and
$\mathopen{}\mathclose{{}\left[\theta}\right]^{1}=\mathopen{}\mathclose{{}\left[\ell}\right]^{0}\mathopen{}\mathclose{{}\left[m}\right]^{0}\mathopen{}\mathclose{{}\left[g}\right]^{0}$.
Thus
$\mathopen{}\mathclose{{}\left[t}\right]^{2}=\mathopen{}\mathclose{{}\left[\ell}\right]\mathopen{}\mathclose{{}\left[g}\right]^{-1}$
and
$\mathopen{}\mathclose{{}\left[\theta}\right]=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$,
so

(5.2) $t^{2}=\ell
g^{-1}\Psi\mathopen{}\mathclose{{}\left(\theta/1_{Q}}\right)=\ell
g^{-1}\Psi\mathopen{}\mathclose{{}\left(\theta}\right).$

Note that if $\kappa=1$ then $\Phi$ is not solvable: there are no integers
$\alpha,\beta,\gamma$ such that
$\mathopen{}\mathclose{{}\left[t}\right]^{1}=\mathopen{}\mathclose{{}\left[\ell}\right]^{\alpha}\mathopen{}\mathclose{{}\left[m}\right]^{\beta}\mathopen{}\mathclose{{}\left[g}\right]^{\gamma}$,
so the canonical $\kappa$ in this case is 2.

Alternatively, choose a local basis $E$ for $\Phi^{*}$ and let $t$, $\ell$,
$g$, $\theta$ denote the scalars
$\mu_{E}\mathopen{}\mathclose{{}\left(t}\right)$,
$\mu_{E}\mathopen{}\mathclose{{}\left(\ell}\right)$,
$\mu_{E}\mathopen{}\mathclose{{}\left(g}\right)$ and
$\mu_{E}\mathopen{}\mathclose{{}\left(\theta}\right)$, respectively, rather
than the corresponding quantities. Then (5.2) becomes $t^{2}=\ell
g^{-1}\psi\mathopen{}\mathclose{{}\left(\theta}\right)$, where
$\psi:\mathbb{R}\rightarrow\mathbb{R}$ is a scaling-covariant scalar
representation of $\Psi$, and from the assumption that $t,\ell,g>0$ follows
that $\psi\mathopen{}\mathclose{{}\left(\theta}\right)>0$ and that (5.2) is
equivalent to

$t=\sqrt{\ell g^{-1}\psi\mathopen{}\mathclose{{}\left(\theta}\right)}.$

It can be shown [18] that
$\sqrt{\psi\mathopen{}\mathclose{{}\left(\theta}\right)}\rightarrow 2\pi$ as
$\theta\rightarrow 0$, so for small oscillations this equation simplifies to
$t=2\pi\sqrt{\ell g^{-1}}$.

######  Example 7.

Let $\mathfrak{A}$ and $\mathfrak{B}$ be two bodies of mass $a$ and $b$,
respectively, let $c$ be the mass of $\mathfrak{A}$ and $\mathfrak{B}$
combined, and assume that we have

$c^{\kappa}=\Phi(a,b).$

The simple dimensional matrix is

$\begin{array}[]{cccc}&\mathopen{}\mathclose{{}\left[c}\right]&\mathopen{}\mathclose{{}\left[a}\right]&\mathopen{}\mathclose{{}\left[b}\right]\\\
\mathsf{M}&1&1&1\end{array},$

and $\mathopen{}\mathclose{{}\left\\{a}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{b}\right\\}$ are the two prebases for $\Phi$,
so if $\kappa=1$ we obtain the equations

$\begin{cases}c^{1}=\Phi_{1}^{*}\mathopen{}\mathclose{{}\left(a}\right)\mathopen{}\mathclose{{}\left(b^{1}}\right),\\\
c^{1}=\Phi_{2}^{*}\mathopen{}\mathclose{{}\left(b}\right)\mathopen{}\mathclose{{}\left(a^{1}}\right).\end{cases}$

where $\mathopen{}\mathclose{{}\left\\{a}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{b}\right\\}$ are local bases for
$\Phi_{1}^{*}$ and $\Phi_{2}^{*}$, respectively. Hence,

$\begin{cases}c=a\,\Psi_{1}\mathopen{}\mathclose{{}\left(b/a}\right),&(M1)\\\
c=b\,\Psi_{2}\mathopen{}\mathclose{{}\left(a/b}\right).&(M2)\end{cases}$

Thus,
$a\Psi_{1}\mathopen{}\mathclose{{}\left(b/a}\right)=b\Psi_{2}\mathopen{}\mathclose{{}\left(a/b}\right)$
for non-zero $a,b\in\mathsf{M}$, and if we assume by symmetry that
$\Phi(a,b)=\Phi(b,a)$ then
$a\Psi_{1}\mathopen{}\mathclose{{}\left(b/a}\right)=a\Psi_{2}\mathopen{}\mathclose{{}\left(b/a}\right)$,
so $\Psi_{1}=\Psi_{2}=\Psi$. Setting $x=b/a$ we thus obtain a functional
equation of the form
$\Psi\mathopen{}\mathclose{{}\left(x}\right)=F\mathopen{}\mathclose{{}\left(x,\Psi}\right)$,
namely

$\Psi\mathopen{}\mathclose{{}\left(x}\right)=x\,\Psi\mathopen{}\mathclose{{}\left(x^{-1}}\right),$

which has solutions of the form

$\Psi(x)=k\mathopen{}\mathclose{{}\left(1_{Q}+x}\right)\qquad\mathopen{}\mathclose{{}\left(k,x\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]}\right),$

so

$c=a\,\Psi\mathopen{}\mathclose{{}\left(b/a}\right)=ak\mathopen{}\mathclose{{}\left(1_{Q}+b/a}\right)=k\mathopen{}\mathclose{{}\left(a+b}\right)\qquad\mathopen{}\mathclose{{}\left(a\neq
0_{\mathsf{M}}}\right).$

If we assume that $\Phi(a,0_{\mathsf{M}})=a$ for all non-zero $a\in\mathsf{M}$
then
$a=a\Psi(0_{\mathsf{\mathopen{}\mathclose{{}\left[1_{Q}}\right]}})=ak\mathopen{}\mathclose{{}\left(1_{Q}+0_{\mathsf{\mathopen{}\mathclose{{}\left[1_{Q}}\right]}}}\right)=ak$,
so $k=1_{Q}$, so we obtain $c=a+b$ as one might expect. If also
$\Phi(0_{\mathsf{M}},0_{\mathsf{M}})=0_{\mathsf{M}}$ then

$c=a+b$

for all $a,b\in\mathsf{M}$ (taking care also of hypothetical negative masses).

We note that while a quantity function $\Phi$ is represented as a product of
quantities in dimensional analysis, such a product can sometimes be rewritten
as a sum of quantities. It is also clear that addition of masses exemplifies a
general principle: if we replace $\mathsf{M}$ by any
$\mathsf{X}\neq\mathopen{}\mathclose{{}\left[1_{Q}}\right]$ then from
$a,b,c\in\mathsf{X}$ and
$\Phi\mathopen{}\mathclose{{}\left(a,b}\right)=\Phi\mathopen{}\mathclose{{}\left(b,a}\right)=c$
we can again derive
$\Phi\mathopen{}\mathclose{{}\left(a,b}\right)=k\mathopen{}\mathclose{{}\left(a+b}\right)$
for at least non-zero $a,b$.

######  Example 8 (based on a problem in Buckingham [4], pp. 358–359).

It is assumed that the energy density $u$ at a fixed point is determined by
the strengths $E$ and $H$ of an electric field $\boldsymbol{\mathbf{E}}$ and a
magnetic $\mathbf{H}$-field, respectively, as well as the permittivity
$\epsilon$ and permeability $\mu$ of the medium, that is,

(5.3) $u^{\kappa}=\Phi\mathopen{}\mathclose{{}\left(E,H,\epsilon,\mu}\right).$

The corresponding dimensional matrix is

$\begin{array}[]{cccccc}&\mathopen{}\mathclose{{}\left[u}\right]&\mathopen{}\mathclose{{}\left[E}\right]&\mathopen{}\mathclose{{}\left[H}\right]&\mathopen{}\mathclose{{}\left[\epsilon}\right]&\mathopen{}\mathclose{{}\left[\mu}\right]\\\
\mathsf{L}&-1&1&-1&-3&1\\\ \mathsf{T}&-2&-3&0&4&-2\\\ \mathsf{M}&1&1&0&-1&1\\\
\mathsf{I}&0&-1&1&2&-2\end{array}.$

This matrix has rank 3. There are four prebases for $\Phi$, namely
$\mathopen{}\mathclose{{}\left\\{E,\epsilon,\mu}\right\\}$,
$\mathopen{}\mathclose{{}\left\\{H,\epsilon,\mu}\right\\}$,
$\mathopen{}\mathclose{{}\left\\{E,H,\epsilon}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{E,H,\mu}\right\\}$, and if $\kappa=1$ a
corresponding set of equations

$\begin{cases}u^{1}=\Phi_{1}^{*}\mathopen{}\mathclose{{}\left(E,\epsilon,\mu}\right)\mathopen{}\mathclose{{}\left(H^{1}}\right),\\\
u^{1}=\Phi_{2}^{*}\mathopen{}\mathclose{{}\left(H,\epsilon,\mu}\right)\mathopen{}\mathclose{{}\left(E^{1}}\right),\\\
u^{1}=\Phi_{3}^{*}\mathopen{}\mathclose{{}\left(E,H,\epsilon}\right)\mathopen{}\mathclose{{}\left(\mu^{1}}\right),\\\
u^{1}=\Phi_{4}^{*}\mathopen{}\mathclose{{}\left(E,H,\mu}\right)\mathopen{}\mathclose{{}\left(\epsilon^{1}}\right),\end{cases}$

where the prebases for $\Phi$ are local bases for
$\Phi_{1}^{*}\text{\textendash}\Phi_{4}^{*}$. Hence, corresponding to
$\Phi_{1}^{*}$– $\Phi_{4}^{*}$,

(5.4)
$\begin{cases}u=E^{2}\epsilon\,\Psi_{1}\mathopen{}\mathclose{{}\left(H^{2}/\mathopen{}\mathclose{{}\left(E^{2}\epsilon\mu^{-1}}\right)}\right),&\mathopen{}\mathclose{{}\left(EM1}\right)\\\
u=H^{2}\mu\,\Psi_{2}\mathopen{}\mathclose{{}\left(E^{2}/\mathopen{}\mathclose{{}\left(H^{2}\epsilon^{-1}\mu}\right)}\right),&\mathopen{}\mathclose{{}\left(EM2}\right)\\\
u=E^{2}\epsilon\,\Psi_{3}\mathopen{}\mathclose{{}\left(\mu/\mathopen{}\mathclose{{}\left(E^{2}H^{-2}\epsilon}\right)}\right),&\mathopen{}\mathclose{{}\left(EM3}\right)\\\
u=H^{2}\mu\,\Psi_{4}\mathopen{}\mathclose{{}\left(\epsilon/\mathopen{}\mathclose{{}\left(E^{-2}H^{2}\mu}\right)}\right).&\mathopen{}\mathclose{{}\left(EM4}\right)\end{cases}$

We note that $E,H,\epsilon,\mu$ occur only in the combinations
$E^{\prime}=\epsilon E^{2}$ and $H^{\prime}=\mu H^{2}$ in these equations.
Also, $\Psi_{1}=\Psi_{3}$ and $\Psi_{2}=\Psi_{4}$, so it suffices to consider
$\Psi_{1}$ and $\Psi_{2}$, say. Thus, we can rewrite (5.4) as

$\begin{cases}u=\epsilon
E^{2}\,\Psi_{1}^{\prime}\mathopen{}\mathclose{{}\left(\mu H^{2}/\epsilon
E^{2}}\right),&(EM1^{\prime}),\\\ u=\mu
H^{2}\,\Psi_{2}^{\prime}\mathopen{}\mathclose{{}\left(\epsilon E^{2}/\mu
H^{2}}\right).&(EM2^{\prime}).\end{cases}$

Buckingham also finds these representations, using the notation
$\varphi_{1}\mathopen{}\mathclose{{}\left(\frac{\mu H^{2}}{\epsilon
E^{2}}}\right)$ and $\varphi_{2}\mathopen{}\mathclose{{}\left(\frac{\epsilon
E^{2}}{\mu H^{2}}}\right)$ [4, p. 359]. He then remarks:

> Assuming that the complete formula is
>
> $u=\frac{1}{8\pi}\mathopen{}\mathclose{{}\left(\epsilon E^{2}+\mu
> H^{2}}\right)$
>
> we have
>
>
> $\varphi_{1}\mathopen{}\mathclose{{}\left(x}\right)=\varphi_{2}\mathopen{}\mathclose{{}\left(x}\right)=\frac{1+x}{8\pi}.$

We have reason to believe that Buckingham already knew “the complete formula”
and then reasoned backwards to obtain
$\varphi_{1}\mathopen{}\mathclose{{}\left(x}\right)=\varphi_{2}\mathopen{}\mathclose{{}\left(x}\right)=\frac{1+x}{8\pi}$.
In dimensional analysis, we are not supposed to know ”the complete formula” at
the outset, however, so let us reverse Buckingham’s inference, proceeding as
in Example 7.

We have
$\mathopen{}\mathclose{{}\left[u}\right]=\mathopen{}\mathclose{{}\left[E^{\prime}}\right]=\mathopen{}\mathclose{{}\left[H^{\prime}}\right]\neq\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
and if we assume for symmetry reasons that
$u=\Phi^{\prime}\mathopen{}\mathclose{{}\left(E^{\prime},H^{\prime}}\right)=\Phi^{\prime}$$\mathopen{}\mathclose{{}\left(H^{\prime},E^{\prime}}\right)$
and set $x=H^{\prime}/E^{\prime}$ then we obtain the functional equation
$\Psi\mathopen{}\mathclose{{}\left(x}\right)=x\,\Psi\mathopen{}\mathclose{{}\left(x^{-1}}\right)$
again. Thus,
$\Psi\mathopen{}\mathclose{{}\left(x}\right)=k\mathopen{}\mathclose{{}\left(1_{Q}+x}\right)$,
where $k,x\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]$, and substitution in
$\mathopen{}\mathclose{{}\left(EM1^{\prime}}\right)$ or
$\mathopen{}\mathclose{{}\left(EM2^{\prime}}\right)$ gives

$u=k\mathopen{}\mathclose{{}\left(\epsilon E^{2}+\mu H^{2}}\right),$

or $u=k\mathopen{}\mathclose{{}\left(\epsilon E^{2}+\mu^{-1}B^{2}}\right)$ if
$B=\mu H$.

Working with examples, Buckingham thus recognised in [4] that there may be
more than one way of representing $\Phi$. However, he dismissed this
observation by asserting that then the representations are ”equivalent” [4, p.
359, 362], implying that it suffices to consider only one of them – any one.
Already this example suggests otherwise; the next example gives more evidence
that Buckingham’s thesis about the equivalence and redundancy of
representations was a mistake, albeit subtle.

######  Example 9 (based on a problem in Bridgman [3], pp. 5–8).

Let two bodies $\mathfrak{B}$ and $\mathfrak{b}$ with masses $M$ and $m$
revolve around each other in empty space under influence of their mutual
gravitational attraction, as in the classical two-body problem. Let $t$ denote
the period of revolution and $d$ the mean distance between $\mathfrak{B}$ and
$\mathfrak{b}$ (or another characteristic distance). One might want to find
out how $t$ depends on $M$, $m$ and $d$, that is,

$t^{\kappa}=\Phi_{0}\mathopen{}\mathclose{{}\left(M,m,d}\right),$

but the associated dimensional matrix

$\begin{array}[]{ccccc}&\mathopen{}\mathclose{{}\left[t}\right]&\mathopen{}\mathclose{{}\left[M}\right]&\mathopen{}\mathclose{{}\left[m}\right]&\mathopen{}\mathclose{{}\left[d}\right]\\\
\mathsf{L}&0&0&0&1\\\ \mathsf{T}&1&0&0&0\\\ \mathsf{M}&0&1&1&0\end{array}$

shows that $\Phi_{0}$ is not precomplete; there is no prebasis of dimensions
for $\Phi_{0}$.

Bridgman suggests that $t$ does also depend on the gravitational constant _G_
, that is,

$t^{\kappa}=\Phi\mathopen{}\mathclose{{}\left(M,m,d,G}\right).$

giving the dimensional matrix

$\begin{array}[]{cccccc}&\mathopen{}\mathclose{{}\left[t}\right]&\mathopen{}\mathclose{{}\left[M}\right]&\mathopen{}\mathclose{{}\left[m}\right]&\mathopen{}\mathclose{{}\left[d}\right]&\mathopen{}\mathclose{{}\left[G}\right]\\\
\mathsf{L}&0&0&0&1&3\\\ \mathsf{T}&1&0&0&0&-2\\\
\mathsf{M}&0&1&1&0&-1\end{array},$

so $\mathopen{}\mathclose{{}\left\\{M,d,G}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{m,d,G}\right\\}$ are the two prebases for
$\Phi$. If $\kappa=2$ we obtain the equations

$\begin{cases}t^{2}=\Phi_{1}^{*}\mathopen{}\mathclose{{}\left(M,d,G}\right)\mathopen{}\mathclose{{}\left(m^{1}}\right),\\\
t^{2}=\Phi_{2}^{*}\mathopen{}\mathclose{{}\left(m,d,G}\right)\mathopen{}\mathclose{{}\left(M^{1}}\right),\end{cases}$

where $\mathopen{}\mathclose{{}\left\\{M,d,G}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{m,d,G}\right\\}$ are local bases for
$\Phi_{1}^{*}$ and $\Phi_{2}^{*}$, respectively. Hence, we have

$\begin{cases}t^{2}=M^{-1}d^{3}G^{-1}\,\Psi_{1}\mathopen{}\mathclose{{}\left(m/M}\right),&(K1)\\\
t^{2}=m^{-1}d^{3}G^{-1}\,\Psi_{2}\mathopen{}\mathclose{{}\left(M/m}\right).&(K2)\end{cases}$

Thus,
$M^{-1}\Psi_{1}\mathopen{}\mathclose{{}\left(m/M}\right)=m^{-1}\Psi_{2}\mathopen{}\mathclose{{}\left(M/m}\right)$,
and for symmetry reasons we assume that
$\Phi\mathopen{}\mathclose{{}\left(M,m,d,G}\right)=\Phi\mathopen{}\mathclose{{}\left(m,M,d,G}\right)$
so that
$M^{-1}\Psi_{1}\mathopen{}\mathclose{{}\left(m/M}\right)=M^{-1}\Psi_{2}\mathopen{}\mathclose{{}\left(m/M}\right)$.
This implies $\Psi_{1}=\Psi_{2}=\Psi$, so setting $x=m/M$ we obtain the
functional equation

$\Psi\mathopen{}\mathclose{{}\left(x}\right)=x^{-1}\,\Psi\mathopen{}\mathclose{{}\left(x^{-1}}\right).$

This functional equation has solutions of the form

$\Psi(x)=k\mathopen{}\mathclose{{}\left(1_{Q}+x}\right)^{-1}\qquad\mathopen{}\mathclose{{}\left(k,x\in\mathopen{}\mathclose{{}\left[1_{Q}}\right]}\right),$

and substituting this in either ($K1$) or ($K2$) gives

(5.5) $t^{2}=kd^{3}G^{-1}(M+m)^{-1}.$

Here, $k$ and $G$ are constants, so if $M+m\approx M$ and $M$ is constant
(several planets orbit the sun) then, approximately, $t^{2}\propto d^{3}$;
this is Kepler’s third law of planetary motion.

As before, (5.5) can be interpreted both as a quantity equation and as a
scalar equation, and assuming that $t,d,G,M,m>0$ the latter can also be
written as

(5.6) $t=c\sqrt{d^{3}G^{-1}(M+m)^{-1}}\qquad\mathopen{}\mathclose{{}\left(c\in
K}\right).$

It is worth pointing out that Bridgman [3, p. 8] considered only one equation,
namely

$t=\frac{r^{\frac{3}{2}}}{G^{\frac{1}{2}}m_{2}^{\frac{1}{2}}}\phi\mathopen{}\mathclose{{}\left(\frac{m_{2}}{m_{1}}}\right),$

corresponding to ($K1$), with
$\phi\mathopen{}\mathclose{{}\left(m_{2}/m_{1}}\right)$ corresponding to
$\Psi_{1}\mathopen{}\mathclose{{}\left(m_{1}/m_{2}}\right)$, or ($K2$), with
$\phi\mathopen{}\mathclose{{}\left(m_{2}/m_{1}}\right)$ corresponding to
$\Psi_{2}\mathopen{}\mathclose{{}\left(m_{1}/m_{2}}\right)$. The basic reason
why Bridgman was not able to derive the much more informative equation (5.6)
was that, unlike Buckingham before him, he did not reflect on the possibility
that the original function could have more than one representation, and as a
consequence he did not reflect on what inferences could be drawn from
symmetries between different representations.

######  Remark 4.

To derive (5.5) in Example 9 we assumed that
$\Phi\mathopen{}\mathclose{{}\left(M,m,d,G}\right)=\Phi\mathopen{}\mathclose{{}\left(m,M,d,G}\right)$
since we may reverse the roles of $\mathfrak{B}$ and $\mathfrak{b}$. This
presupposes that a description of the two-body system in which $\mathfrak{B}$
is at rest and $\mathfrak{b}$ revolves around $\mathfrak{B}$ is empirically
equivalent to a description in which $\mathfrak{b}$ is at rest and
$\mathfrak{B}$ revolves around $\mathfrak{b}$, meaning that only relative
motion can be observed. If there are empirical observations inconsistent with
this symmetry assumption, there is no ground for assuming that
$\Phi\mathopen{}\mathclose{{}\left(M,m,d,G}\right)=\Phi\mathopen{}\mathclose{{}\left(m,M,d,G}\right)$.
In general, dimensional analysis aimed at deriving physical laws is ultimately
based on empirically testable assumptions [12].

##  6\. Dimensional analysis and matroid theory

A dimensional matrix can be analyzed from the point of view of matroid theory.
Recall that a (finite) matroid is a finite set $\mathcal{E}$ equipped with a
set $\mathcal{I}$ of subsets of $\mathcal{E}$, said to be _independent_ sets;
$\mathcal{I}$ is required to satisfy certain conditions so as to generalize
the notion of (linear) independence in a vector space, a free abelian group
etc.

A set $S=\mathopen{}\mathclose{{}\left\\{q_{1},\ldots,q_{p}}\right\\}$ of
quantity variables, corresponding to dimensions
$\mathopen{}\mathclose{{}\left[q_{1}}\right],\ldots,\mathopen{}\mathclose{{}\left[q_{p}}\right]$
in $Q/{\sim}$, is a matroid with its _maximal independent sets_ of variables
in $S$ reflecting the maximal sets of linearly independent associated columns
in an $m\times p$ dimensional matrix. Such a maximal independent set, known as
a _basis_ in matroid theory, corresponds to a _“_ group of repeating variables
_”_ in traditional dimensional analysis (see, e.g., [21, Section 5.2]), and a
prebasis for some quantity function $\Phi$ on $Q$ in the present approach.

Those subsets of $\mathcal{E}$ which are not independent are said to be
_dependent_. A minimal dependent set of variables in $S$ is called a _circuit_
in matroid theory. A _pseudocircuit_ is a subset $C$ of $\mathcal{E}$ such
that $C\setminus\mathopen{}\mathclose{{}\left\\{e}\right\\}$ is a basis for
some $e\in C$. A pseudocircuit corresponds to a “dimensionless group” or _“_
Pi group _”_ in traditional dimensional analysis.

Every matroid basis contains the same number of variables, which is the rank
$r$ of the dimensional matrix, so every pseudocircuit contains exactly $r+1$
variables. Hence, a pseudocircuit is a set of $r+1$ variables with a subset
that is a basis. For an $m\times p$ dimensional matrix defining a matroid
there are at most $\binom{p}{r}$ bases and $\binom{p}{r+1}$ pseudocircuits,
where $r\leq m,p-1$.

######  Example 10.

Consider the dimensional matrix

$\begin{array}[]{cccc}&\mathopen{}\mathclose{{}\left[a}\right]&\mathopen{}\mathclose{{}\left[b}\right]&\mathopen{}\mathclose{{}\left[c}\right]\\\
\mathsf{E}_{1}&1&1&1\\\ \mathsf{E}_{2}&0&1&1\end{array}.$

Here, $\mathopen{}\mathclose{{}\left\\{a,b}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{a,c}\right\\}$ are bases,
$\mathopen{}\mathclose{{}\left\\{a,b,c}\right\\}$ is a pseudocircuit but not a
circuit, and $\mathopen{}\mathclose{{}\left\\{b,c}\right\\}$ is a circuit but
not a pseudocircuit.

For every pseudocircuit, there are integers $k_{j}$, unique up to sign, that
define a pair of $\pi$_-monomials_ of the form $x_{1}^{k_{1}}\cdots
x_{r+1}^{k_{r+1}}$ such that

(6.1)
$\prod_{j=1}^{r+1}\nolimits\mathopen{}\mathclose{{}\left[x_{j}}\right]^{k_{j}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right],\qquad\exists
j:k_{j}\neq
0,\qquad\mathrm{gcd}\mathopen{}\mathclose{{}\left(k_{1},\ldots,k_{r+1}}\right)=1,$

where $x_{1},\ldots,x_{r+1}$ are the variables in the pseudocircuit.
(Uniqueness follows from the rank-nullity theorem for free
$\mathbb{Z}$-modules, according to which the kernel of an
$m\times\mathopen{}\mathclose{{}\left(r+1}\right)$ integer matrix of rank $r$
has rank 1.) Note that a “dimensionless” monomial $q_{1}^{k_{1}}\cdots
q_{p}^{k_{p}}$ satisfying
$\prod_{j=1}^{p}\mathopen{}\mathclose{{}\left[q_{j}}\right]^{k_{j}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
rather than the stronger condition
$\prod_{j=1}^{r+1}\nolimits\mathopen{}\mathclose{{}\left[x_{j}}\right]^{k_{j}}=\mathopen{}\mathclose{{}\left[1_{Q}}\right]$
need not be a $\pi$-monomial, and that the proof of Theorem 1 requires that
the variables $\pi_{0},\ldots,\pi_{n-r}$ are $\pi$-monomials as defined here.

######  Example 11.

Recall that the structure of a matroid is fully specified by its set of bases.
This set, and the set of pseudocircuits, can be exhibited in a compact form as
a matrix. The dimensional matrix in Example 9 generates a matroid
$\mathcal{M}$, represented below:

(6.2)
$\begin{array}[]{ccccccccccccc}&A&B&C&D&E&F&G&\alpha&\beta&\gamma&\delta&\epsilon\\\
t&+&+&+&+&+&-&-&\ast&\ast&\ast&\ast&\circ\\\
M&+&+&-&-&-&+&-&\ast&\ast&\ast&\circ&\ast\\\
m&-&-&+&+&-&-&+&\ast&\ast&\circ&\ast&\ast\\\
d&+&-&+&-&+&+&+&\ast&\circ&\ast&\ast&\ast\\\
G&-&+&-&+&+&+&+&\circ&\ast&\ast&\ast&\ast\end{array}.$

Each column labelled with a capital letter specifies a basis in $\mathcal{M}$,
containing variables with a plus sign in this column. Also, each column
labelled by a Greek letter specifies a pseudocircuit, containing variables
with an asterisk in this column. For example, Column $A$ in the table above
specifies the basis $\mathopen{}\mathclose{{}\left\\{t,M,d}\right\\}$,
contained in the pseudocircuits
$\alpha=\mathopen{}\mathclose{{}\left\\{t,M,m,d}\right\\}$ and
$\gamma=\mathopen{}\mathclose{{}\left\\{t,M,d,G}\right\\}$.

Corresponding to the pseudocircuits

$\alpha=\mathopen{}\mathclose{{}\left\\{t,M,m,d}\right\\},\quad\beta=\mathopen{}\mathclose{{}\left\\{t,M,m,G}\right\\},\quad\mathfrak{\gamma}=\mathopen{}\mathclose{{}\left\\{t,M,d,G}\right\\},\quad\mathfrak{\delta}=\mathopen{}\mathclose{{}\left\\{t,m,d,G}\right\\},\quad\epsilon=\mathopen{}\mathclose{{}\left\\{M,m,d,G}\right\\}$

we have the $\pi$_-_ monomial pairs
$\mathopen{}\mathclose{{}\left\\{\pi_{a},\pi_{a}^{-1}}\right\\}$,
$\mathopen{}\mathclose{{}\left\\{\pi_{\gamma},\pi_{\gamma}^{-1}}\right\\}$,
$\mathopen{}\mathclose{{}\left\\{\pi_{\delta},\pi_{\delta}^{-1}}\right\\}$,
where

$\pi_{\alpha}=\pi_{\beta}=\pi_{\epsilon}=Mm^{-1},\quad\pi_{\gamma}=t^{2}Md^{-3}G,\quad\pi_{\delta}=t^{2}md^{-3}G,$

omitting variables with exponent $0$, keeping the original order of the
variables and letting the exponent of the first variable be positive. (Note
that
$\alpha\cap\beta\cap\epsilon=\mathopen{}\mathclose{{}\left\\{M,m}\right\\}$.)

As suggested by Vaschy [20] and Buckingham [4], we may let the dependencies
among $t,M,m,d,G$ be described by equations of the form
$F\mathopen{}\mathclose{{}\left(\pi_{1},\ldots,\pi_{n}}\right)=0$. Combining
$\pi$-monomials into equations, we require that $\pi$-monomials in the same
equation correspond to pseudocircuits containing the same basis. The
$\pi$-monomials above then generate the equations
$F\mathopen{}\mathclose{{}\left(\pi_{\alpha},\pi_{\gamma}}\right)=0_{\mathopen{}\mathclose{{}\left[1_{Q}}\right]}$,
$F\mathopen{}\mathclose{{}\left(\pi_{\alpha},\pi_{\delta}}\right)=0_{\mathopen{}\mathclose{{}\left[1_{Q}}\right]}$
and
$F\mathopen{}\mathclose{{}\left(\pi_{\gamma},\pi_{\delta}}\right)=0_{\mathopen{}\mathclose{{}\left[1_{Q}}\right]}$
since $\alpha\cap\gamma=A$, $\alpha\cap\delta=C$ and $\gamma\cap\delta=E$,
respectively.

We mostly want to write our derived equations in a more explicit form with a
single dependent variable, however. Then we need an additional restriction:
the basis associated with an equation must not contain the dependent variable.
Let us continue to review Example 9 from this point of view.

For easy reference, we show the equation system at the center of Example 9
again:

$\begin{cases}t^{2}=M^{-1}d^{3}G^{-1}\,\Psi_{1}\mathopen{}\mathclose{{}\left(m/M}\right),\\\
t^{2}=m^{-1}d^{3}G^{-1}\,\Psi_{2}\mathopen{}\mathclose{{}\left(M/m}\right).\end{cases}$

We note that the three $\pi$-monomial pairs are all used in these two
equations, but there are four more possible equation systems that also use
these $\pi$-monomial pairs as building blocks:

$\begin{cases}M=m\Psi_{11}\mathopen{}\mathclose{{}\left(G/t^{-2}m^{-1}d^{3}}\right),\\\
M=m\Psi_{12}\mathopen{}\mathclose{{}\left(t^{2}/m^{-1}d^{3}G^{-1}}\right)\quad\mathopen{}\mathclose{{}\left(t\geq
0_{\mathopen{}\mathclose{{}\left[t}\right]}}\right),\\\
M=m\Psi_{13}\mathopen{}\mathclose{{}\left(d^{3}/t^{2}mG}\right),\\\
M=t^{-2}d^{3}G^{-1}\Psi_{14}\mathopen{}\mathclose{{}\left(m/t^{-2}d^{3}G^{-1}}\right).\end{cases}\qquad\begin{cases}m=M\Psi_{21}\mathopen{}\mathclose{{}\left(G/t^{-2}M^{-1}d^{3}}\right),\\\
m=M\Psi_{22}\mathopen{}\mathclose{{}\left(t^{2}/M^{-1}d^{3}G^{-1}}\right)\quad\mathopen{}\mathclose{{}\left(t\geq
0_{\mathopen{}\mathclose{{}\left[t}\right]}}\right),\\\
m=M\Psi_{23}\mathopen{}\mathclose{{}\left(d^{3}/t^{2}MG}\right),\\\
m=t^{-2}d^{3}G^{-1}\Psi_{24}\mathopen{}\mathclose{{}\left(M/t^{-2}d^{3}G^{-1}}\right).\end{cases}$
$\begin{cases}d^{3}=t^{2}Mg\Psi_{31}\mathopen{}\mathclose{{}\left(m/M}\right),\\\
d^{3}=t^{2}mG\Psi_{32}\mathopen{}\mathclose{{}\left(M/m}\right).\end{cases}\qquad\begin{cases}G=t^{-2}M^{-1}d^{3}\Psi_{41}\mathopen{}\mathclose{{}\left(m/M}\right),\\\
G=t^{-2}m^{-1}d^{3}\Psi_{42}\mathopen{}\mathclose{{}\left(M/m}\right).\end{cases}$

Thus, there is one equation system of the form (4.5) for each variable $V,$
and each equation in that equation system corresponds to a column in (6.2)
marked by a capital letter and with a “$-$” in the row where $V$ occurs. For
example, $\gamma\cap\epsilon=F$, and the equation
$t^{2}Md^{-3}G=\Psi_{1}\mathopen{}\mathclose{{}\left(M^{-1}m}\right)$, or
equivalently
$t^{2}=M^{-1}d^{3}G^{-1}\,\Psi_{1}\mathopen{}\mathclose{{}\left(m/M}\right)$,
is associated with the basis $F$ via the $\pi$ -monomials $\pi_{\gamma}$ and
$\pi_{\alpha}^{-1}$, whereas $\delta\cap\epsilon=G$, and
$t^{2}md^{-3}G=\psi\mathopen{}\mathclose{{}\left(Mm^{-1}}\right)$, or
equivalently
$t^{2}=m^{-1}d^{3}G^{-1}\,\Psi_{2}\mathopen{}\mathclose{{}\left(M/m}\right),$
is associated with $G$ via the $\pi$-monomials $\pi_{\delta}$ and
$\pi_{\alpha}$.

In the approach to dimensional analysis described in this section, the choice
of a dependent variable (if any) is suspended; this is a “balanced” or
“explorative” approach. It should be noted, however, that a subsequent choice
of a particular variable in a dimensional matrix as the dependent variable may
be more or less meaningful, depending on the physical (rather than
mathematical) nature of the problem analyzed as well as the information
sought.

######  Remark 5.

The method to construct equations illustrated in Example 11 leads to some
redundancy. For example, we must have $\Psi_{11}=\Psi_{12}$ when $t\geq
0_{\mathopen{}\mathclose{{}\left[t}\right]}$ since
$G/t^{-2}m^{-1}d^{3}=t^{2}/m^{-1}d^{3}G^{-1}$. There are two equations rather
than one because the pseudocircuit
$\mathopen{}\mathclose{{}\left\\{t,m,d,G}\right\\}$ contains the two bases
$\mathopen{}\mathclose{{}\left\\{t,m,d}\right\\}$ and
$\mathopen{}\mathclose{{}\left\\{m,d,G}\right\\}$, and for each equation
system there is one equation for each basis. We can disregard all but one of
such duplicated equations, however; here it suffices to use $\Psi_{11}$.

##  7\. A remark on covariance of scalar representations

Dimensional analysis is based on the principle of covariance: a relation
between scalars representing a relation between quantities relative to a
reference frame must continue to hold when the reference frame is changed. The
reference frame in this case is a basis for a quantity space.

A “physically meaningful” quantity equation $\Phi$ is precisely a complete
quantity function with a covariant scalar representation, but in Theorems 1
and 2 scalar representations of quantity functions are only required to be
scaling-covariant. It is plausible that it would be possible to derive
stronger results if general covariance was assumed, making full use of the
postulate that $\Phi$ is “physically meaningful”, but it is not clear how such
stronger versions of Theorems 1 and 2 should be formulated. On the other hand,
it may be argued that as we can apply the two theorems separately to each
$\Phi_{i}^{\prime}$ or $\Phi_{i}^{*}$ and then combine the results, we have a
way to make use of a more general kind of covariance than that underlying
traditional dimensional analysis.

Recall that in the traditional approach to dimensional analysis, all
dimensions are expressed in terms of a fixed dimensional basis for $Q/{\sim}$
such as
$\mathopen{}\mathclose{{}\left\\{\mathsf{L},\mathsf{M},\mathsf{T}}\right\\}$.
Thus, any change of measures of quantities are obtained by a rescaling of a
corresponding system of units of measurement, for example
$\mathopen{}\mathclose{{}\left\\{\mathrm{m},\mathrm{kg},\mathrm{s}}\right\\}$.
In the present approach, the same is true for each local dimensional basis,
but there is in general more than one local dimensional basis involved,
meaning that more information may be provided by the assumption of scaling-
covariance, and as a result more information can often be obtained from
dimensional analysis, as shown by examples in Section 5.

The fact that dimensional analysis may yield more than one possible equation
has previously – when it has not been overlooked – mostly been regarded as a
“non-uniqueness” problem [21, p. 300], dealt with by arbitrarily choosing one
equation or by introducing pragmatic rules to single out the “right” equation.
This “non-uniqueness” could instead be seen as a useful result of the
extension of scaling-covariance towards general covariance, thereby allowing
dimensional analysis to yield an entire system of equations in the unbalanced
case and a set of systems of equations in the balanced case, opening up new
prospects for dimensional analysis.

## References

  * [1] de Boer, J. (1994). On the history of quantity calculus and the international system, _Metrologia_ , 31, 405–429.
  * [2] Brand, L. (1957). The Pi Theorem of Dimensional Analysis. _Archive for Rational Mechanics and Analysis_ , 1 (1), 35-45.
  * [3] Bridgman P.W. (1922). _Dimensional Analysis_. Yale University Press.
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##  Appendix A A local basis is a basis

Let $Q$ be a quantity space,
$\Phi\mathopen{}\mathclose{{}\left(q_{1},\ldots,q_{n}}\right)\mapsto q_{0}$ a
quantity function on $Q$,
$\mathopen{}\mathclose{{}\left\\{e_{1},\ldots,e_{r}}\right\\}$ a local basis
for $\Phi$, $Q^{\prime}$ the subgroup of $Q$ generated by
$q_{0},\ldots,q_{n}$, and $q\in Q^{\prime}$. If
$q=\mu\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{j}}=\mu^{\prime}\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{j}^{\prime}}$ then $\kappa_{j}=\kappa_{j}^{\prime}$ since
$\mathopen{}\mathclose{{}\left[q}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{j}}}\right]$ and $1,\kappa_{1},\ldots,\kappa_{r}$ are unique
canonical exponents. We also have $\mu=\mu^{\prime}$ since
$\prod_{j=1}^{r}\nolimits e_{j}^{\kappa_{j}}$ is non-zero. Thus, the expansion

(A.1)
$q=\mu_{E}\mathopen{}\mathclose{{}\left(q}\right)\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{j}}$

is unique if it exists.

Furthermore, there are unique integers $\kappa_{ij}$ such that
$\mathopen{}\mathclose{{}\left[q_{i}}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}}\right]$
for $i=0,\ldots,n$. If
$E_{Q}=\mathopen{}\mathclose{{}\left\\{\overline{e}_{1},\ldots,\overline{e}_{m}}\right\\}$
is a basis for $Q$ then
$q_{i}=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(q_{i}}\right)\cdot\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}}$
and
$\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}}\right)\cdot\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}^{\prime}}$.
Here, $\ell_{ij}=\ell_{ij}^{\prime}$ for $j=1,\ldots,m$ since
$\mathopen{}\mathclose{{}\left[q_{i}}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}}\right]$
implies
$\mathopen{}\mathclose{{}\left[\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}}}\right]=\mathopen{}\mathclose{{}\left[\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}^{\prime}}}\right]$,
so that
$x=\alpha\cdot\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}}=\beta\cdot\prod_{j=1}^{m}\overline{e}_{j}^{\ell_{ij}^{\prime}}$,
where the expansion of $x$ is unique. Also,
$\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}}\right)\neq
0$ since $\prod_{j=1}^{r}e_{j}^{\kappa_{ij}}$ is non-zero. Hence,

$\displaystyle q_{i}$
$\displaystyle=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(q_{i}}\right)\cdot\prod\nolimits_{j=1}^{m}\nolimits\overline{e}_{j}^{\ell_{ij}}=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(q_{i}}\right)\cdot\mathopen{}\mathclose{{}\left(\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(\prod\nolimits_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}}}\right)^{-1}\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}}}\right)\cdot\prod_{j=1}^{m}\nolimits\overline{e}_{j}^{\ell_{ij}^{\prime}}}\right)$
$\displaystyle=\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(q_{i}}\right)\,\mu_{E_{Q}}\mathopen{}\mathclose{{}\left(\prod\nolimits_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}}}\right)^{-1}\cdot\prod_{j=1}^{r}\nolimits
e_{j}^{\kappa_{ij}},$

and by assumption $q=\mu\cdot\prod_{j=0}^{n}q_{j}^{k_{j}}$ for some $\mu$ and
$k_{j}$, so an expansion of the form (A.1) exists.