Namely, let $\ell_1$ and $\ell_2$ be any curves in  $\mathcal{Q}=\mathbb{P}^1\times\mathbb{P}^1$ of degrees $(1,0)$ and $(0,1)$, respectively.
For $n\in\mathbb{Z}_{\geqslant 0}$ and  $i\in\{2,3,4,5,6,7,8\}$, let
$B_{n,1,1}$, $B_{n,1,2}$, $B_{n,2,i}$, $B_{n,3}$, $B_{n,4,i}$ be the classes

\varpi^*\big(a_1\ell_1+a_2\ell_2\big)-\sum_{i=1}^8b_i\mathbf{e}_i\in\mathrm{Pic}(T),

where $a_1,a_2,b_1,\ldots,b_8$ are non-negative integers given in the following table:

$a_1$     $a_2$     $b_1$     $b_2,b_3,b_4,b_5,b_6,b_7,b_8$

$B_{n,1,1}$     $14n^{2}+7n+1$     $14n^{2}+7n$     $7n^{2}+7n+1$     $\forall j$ $b_j=7n^{2}+3n$

$B_{n,1,2}$     $14n^{2}+7n$     $14n^{2}+7n+1$     $7n^{2}+7n+1$     $\forall j$ $b_j=7n^{2}+3n$

$b_i=7 n^{2}+6n+2$
$\forall j\ne i$ $b_j=7n^{2}+6n+1$

$B_{n,3}$     $14n^{2}+21n+7$     $14n^{2}+21 n+7$     $7n^{2}+14 n+6$     $\forall j$ $b_j=7 n^{2}+10n+3$

$14 n^{2}+29 n+15$

$14n^{2}+29 n+15$

$\forall j\ne i$ $b_j=7n^{2}+14n+7$

By Corollary <ref>, each linear system
$|B_{n,1,1}|$, $|B_{n,1,2}|$, $|B_{n,2,i}|$, $|B_{n,3}|$, $|B_{n,4,i}|$ contains a unique $(-1)$-curve.
Hence, we can identify the classes $B_{n,1,1}$, $B_{n,1,2}$, $B_{n,2,i}$, $B_{n,3}$, $B_{n,4,i}$ with $(-1)$-curves
in $|B_{n,1,1}|$, $|B_{n,1,2}|$, $|B_{n,2,i}|$, $|B_{n,3}|$, $|B_{n,4,i}|$, respectively.

\begin{align*}
B_{n, 1}&=B_{n, 1, 1}+B_{n, 1, 2},\\
B_{n, 2}&=B_{n,2,2}+B_{n,2,3}+B_{n,2,4}+B_{n,2,5}+B_{n,2,6}+B_{n,2,7}+B_{n,2,8},\\
B_{n, 4}&=B_{n,4,2}+B_{n,4,3}+B_{n,4,4}+B_{n,4,5}+B_{n,4,6}+B_{n,4,7}+B_{n,4,8}.
\end{align*}

Note that irreducible components of each curve $B_{n,1}$, $B_{n,2}$, $B_{n,4}$ are disjoint $(-1)$-curves,
and $B_{n,1}\cap B_{n,2}=\varnothing$, $B_{n,2}\cap B_{n,3}=\varnothing$, $B_{n,3}\cap B_{n,4}=\varnothing$, $B_{n,4} \cap B_{n+1,1}=\varnothing$ for each $n\geqslant 0$.

Now, we let $I^\prime_{0,1}=[0,\frac{1}{3}]$ and $I^{\prime\prime}_{0,1}=[\frac{1}{3},\frac{3}{8}]$.
For every $n\in\mathbb{Z}_{> 0}$, we also let

\begin{align*}
I_{n, 1}^{\prime}&=\left[\frac{-1+4 n+14 n^{2}}{6 n+14 n^{2}}, \frac{1+13 n+21 n^{2}}{3+16 n+21 n^{2}}\right],\\
I_{n, 1}^{\prime \prime}&=\left[\frac{1+13n+21n^{2}}{3+16n+21n^{2}}, \frac{3+35n+49 n^{2}}{8+42 n+49 n^{2}}\right].
\end{align*}

For every $n\in\mathbb{Z}_{\geqslant 0}$, we let

\begin{align*}
I_{n,2}^{\prime}&=\left[\frac{3+35 n+49n^{2}}{8+42 n+49 n^{2}}, \frac{3+22 n+28 n^{2}}{6+26 n+28 n^{2}}\right],\\
I_{n,2}^{\prime\prime}&=\left[\frac{3+22n+28 n^{2}}{6+26 n+28 n^{2}}, \frac{2+7 n}{3+7 n}\right],\\
I_{n,3}^{\prime}&=\left[\frac{2+7 n}{3+7 n}, \frac{21+50 n+28 n^{2}}{26+54 n+28 n^{2}}\right],\\
I_{n, 3}^{\prime\prime}&=\left[\frac{21+50 n+28 n^{2}}{26+54 n+28 n^{2}}, \frac{39+91 n+49 n^{2}}{48+98 n+49 n^{2}}\right],\\
I_{n,4}^{\prime}&=\left[\frac{39+91 n+49 n^{2}}{48+98 n+49 n^{2}}, \frac{19+41 n+21 n^{2}}{23+44 n+21 n^{2}}\right],\\
I_{n,4}^{\prime\prime}&=\left[\frac{19+41 n+21 n^{2}}{23+44 n+21 n^{2}}, \frac{17+32 n+14 n^{2}}{20+34 n+14 n^{2}}\right].
\end{align*}

Set $I_{n,1}=I_{n,1}^\prime\cup I_{n,1}^{\prime\prime}$, $I_{n,2}=I_{n,2}^\prime\cup I_{n,2}^{\prime\prime}$, $I_{n,3}=I_{n,3}^\prime\cup I_{n,3}^{\prime\prime}$, $I_{n,4}=I_{n,4}^\prime\cup I_{n,4}^{\prime\prime}$.

[0,1)=\bigcup_{n \in \mathbb{Z}_{\geqslant 0}} \Big(I_{n, 1} \cup I_{n, 2} \cup I_{n, 3} \cup I_{n, 4}\Big),

the intervals $I_{n,1}$, $I_{n,2}$, $I_{n,3}$, $I_{n,4}$ have positive volumes, and all their interiors are disjoint.
Let us analyze $P(u,v)$ and $N(u,v)$ when $u$ is contained in one of these intervals.

First, we deal with $u\in I_{n,1}$.
If $u\in I_{n,1}^{\prime}$ and $v\in\left[0,\frac{2+14 n+28 n^{2}-u(1+14 n+28 n^{2})}{1+7 n+7 n^{2}}\right]$,

\begin{multline*}
P(u, v)=\frac{19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)}{8+28 n+21 n^{2}}\mathbf{e}_{1}+\\
+\frac{3+35n+49 n^{2}-u\left(8+42 n+49 n^{2}\right)}{8+28 n+21 n^{2}}B_{n, 1}+
\frac{1-4n-14n^{2}+u\left(6 n+14 n^{2}\right)}{8+28 n+21 n^{2}}B_{n, 2}
\end{multline*}

and $N(u,v)=0$. The same holds if $u\in I_{n,1}^{\prime\prime}$ and .

\begin{align*}
\big(P(u,v)\big)^{2}&=10-12u+2u^{2}-2v-v^{2},\\
\end{align*}

Similarly, if $u\in I_{n,1}^\prime$ and $v\in\left[\frac{2+14 n+28 n^{2}-u\left(1+14 n+28 n^{2}\right)}{1+7 n+7 n^{2}}, \frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u, v)=\frac{19+70n+84n^{2}-u(16+70 n+84 n^{2})-v(8+28 n+21 n^{2})}{8+28 n+21 n^{2}}\mathbf{e}_{1}+\\
+\frac{\big(19+70n+84n^{2}-u(16+70 n+84 n^{2})-v(8+28 n+21 n^{2})\big)(1+7n+7n^{2})}{8+28n+21n^{2}}B_{n,1}+\\
+\frac{1-4n-14n^{2}+u(6 n+14 n^{2})}{8+28 n+21 n^{2}}B_{n,2}\quad\quad\quad\quad\quad\quad
\end{multline*}

N(u,v)=\left(u\left(1+14 n+28 n^{2}\right)+v\left(1+7n+7n^{2}\right)-2-14n-28n^{2}\right)B_{n,1}.

\begin{multline*}
\quad\quad \big(P(u,v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+2\left(u\left(1+14 n+28 n^{2}\right)+v\left(1+7 n+7 n^{2}\right)-2-14n-28n^{2}\right)^{2}\quad\quad\quad\quad
\end{multline*}

\begin{multline*}
\quad\quad P(u,v)\cdot\mathbf{e}_1=5+56n+280n^2+588n^3+392n^4-\\
\quad\quad-2u(1+7n+7n^2)(1+14n+28n^2)-v(1+28n+126n^2+196n^3+98n^4).
\end{multline*}

Likewise, if $u\in I_{n,1}^{\prime\prime}$ and $v \in\left[\frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}, \frac{2+14 n+28 n^{2}-u\left(1+14 n+28 n^{2}\right)}{1+7 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u, v)=\frac{(19+70 n+84 n^{2}-u(16+70 n+84 n^{2})-v(8+28 n+21 n^{2})}{8+28 n+21 n^{2}}\mathbf{e}_{1}+\\
+\frac{\big(19+70 n+84 n^{2}-u(16+70 n+84 n^{2})-v(8+28 n+21 n^{2})\big)(1+n)(3+7n)}{8+28 n+21 n^{2}}B_{n,2}+\\
+\frac{3+35 n+49 n^{2}-u\left(8+42 n+49 n^{2}\right)}{8+28 n+21 n^{2}}B_{n,1}\quad\quad\quad\quad
\end{multline*}

N(u,v)=\left(u\left(6+26 n+28 n^{2}\right)+v\left(3+10 n+7 n^{2}\right)-7-26n-28n^{2}\right)B_{n,2}.

\begin{multline*}
\big(P(u,v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+7\left(u\left(6+26 n+28 n^{2}\right)+v\left(3+10 n+7 n^{2}\right)-7-26-28n^{2}\right)^{2}\quad\quad
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=148+1036 n+2751 n^{2}+3234 n^{3}+1372 n^{4}-\\
-14u(1+n)(1+2 n)(3+7 n)^{2}-v\left(62+420 n+994 n^{2}+980 n^{3}+343 n^{4}\right).\quad\quad
\end{multline*}

If $u \in I_{n, 1}^{\prime}$ and $v\in\left[\frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}, \frac{19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)}{8+28 n+21 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)}{8+28 n+21 n^{2}}\mathbf{e}_1+\\
+\frac{\big(19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)\big)\left(1+7n+7 n^{2}\right)}{8+28 n+21 n^{2}}B_{n,1}+\\
+\frac{\big(19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)\big)(1+n)(3+7n)}{8+28 n+21 n^{2}}B_{n,2}
\end{multline*}

\begin{multline*}
N(u,v)=\left(u\left(1+14 n+28 n^{2}\right)+v\left(1+7 n+7 n^{2}\right)-2-14n-28n^{2}\right)B_{n,1}+ \\
+\left(u\left(6+26 n+28 n^{2}\right)+v\left(3+10 n+7 n^{2}\right)-7-26n-28n^{2}\right)B_{n,2}.
\end{multline*}

The same holds if $u\in I_{n,1}^{\prime\prime}$ and $v\in\left[\frac{2+14 n+28 n^{2}-u\left(1+14 n+28 n^{2}\right)}{1+7 n+7 n^{2}}, \frac{19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)}{8+28 n+21 n^{2}}\right]$.
In both cases, we have

\begin{align*}
\big(P(u,v)\big)^{2}&=\left(19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)\right)^{2},\\
P(u,v)\cdot\mathbf{e}_1&=\left(8+28 n+21 n^{2}\right)\left(19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)-v\left(8+28 n+21 n^{2}\right)\right).
\end{align*}

Hence, if $u\in I_{n,1}$, then

t(u)=\frac{19+70 n+84 n^{2}-u\left(16+70 n+84 n^{2}\right)}{8+28 n+21 n^{2}}.

Now, we deal with $u\in I_{n,2}$. If $u\in I_{n,2}^{\prime}$ and $v\in\left[0,\frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{17+56 n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2 n)}{7(1+n)(1+2n)}\mathbf{e}_{1}+ \\
+\frac{(1+n)(2+7 n)-u(1+n)(3+7n)}{7(1+n)(1+2n)} B_{n, 2}+\frac{u\left(8+42 n+49 n^{2}\right)-3-35n-49n^{2}}{7(1+n)(1+2n)}B_{n, 3}
\end{multline*}

and $N(u,v)=0$.
The same holds if $u\in I_{n,2}^{\prime\prime}$ and $v\in\left[0,\frac{15+42n+28n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}\right]$.

\begin{align*}
\big(P(u,v)\big)^{2}&=10-12u+2u^{2}-2v-v^{2},\\
\end{align*}

If $u\in I_{n,2}^\prime$ and $v \in\left[\frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}, \frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}\right]$, then

\begin{multline*}
\quad\quad P(u,v)=\frac{17+56n+56 n^{2}-u(15+56 n+56 n^{2})-7v(1+n)(1+2n)}{7(1+n)(1+2n)}\mathbf{e}_{1}+\\
+\frac{(1+n)(3+7n)\big(17+56n+56 n^{2}-u(15+56 n+56 n^{2})-7v(1+n)(1+2n)\big)}{7(1+n)(1+2n)}B_{n, 2}+\\
+\frac{u\left(8+42 n+49 n^{2}\right)-3-35n-49n^{2}}{7(1+n)(1+2 n)}B_{n,3},\quad\quad\quad\quad\quad\quad
\end{multline*}

N(u,v)=\big(u\left(6+26 n+28 n^{2}\right)+v\left(3+10 n+7n^{2}\right)-7-26n-28n^{2}\big)B_{n, 2},

\begin{multline*}
\big(P(u,v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+7\left(u\left(6+26 n+28 n^{2}\right)+v\left(3+10 n+7 n^{2}\right)-7-26n-28n^{2}\right)^{2},
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=148+1036 n+2751 n^{2}+3234 n^{3}+1372 n^{4}-\\
\quad\quad\quad-14u(1+n)(1+2 n)(3+7 n)^{2}-\\
-v\left(62+420 n+994 n^{2}+980 n^{3}+343 n^{4}\right).
\end{multline*}

Similarly, if $u\in I_{n,2}^{\prime\prime}$ and $v\in\left[\frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}, \frac{7+26 n+28 n^{3}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{17+56+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2 n)}{7(1+n)(1+2n)}\mathbf{e}_{1}+\\
+\frac{(6+14 n+7 n^{2})\big(17+56+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2 n)\big)}{7(1+n)(1+2 n)}B_{n, 3}+\\
\end{multline*}

\begin{multline*}
\big(P(u, v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+\left(u\left(14+42 n+28 n^{2}\right)+v\left(6+14 n+7 n^{2}\right)-15-42n-28n^{2}\right)^{2},
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=7(1+n)(13+53 n+70n^{2}+28 n^{3})-\\
-14u(1+n)(1+2n)\left(6+14 n+7 n^{2}\right)-7v(1+n)^2\left(5+14 n+7 n^{2}\right).
\end{multline*}

Likewise, if $u\in I_{n,2}^{\prime}$ and $v\in\left[\frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}, \frac{17+56 n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)}{7(1+n)(1+2 n)}\right]$, then

\begin{multline*}
P(u,v)=\frac{17+56n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-v(7(1+n)(1+2 n))}{7(1+n)(1+2n)}\mathbf{e}_{1}+\\
+\frac{(1+n)(3+7n)\big(17+56n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2 n)\big)}{7(1+n)(1+2n)}B_{n, 2}+\\
+\frac{\left(6+14 n+7 n^{2}\right)\big(17+56n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2n)\big)}{7(1+n)(1+2n)}B_{n,3}
\end{multline*}

\begin{multline*}
N(u,v)=\big(u(6+26 n+28 n^{2})+v(3+10 n+7 n^{2})-7-26n-28n^{2}\big) B_{n, 2}+\\
+\big(u\left(14+42 n+28 n^{2}\right)+v\left(6+14 n+7 n^{2}\right)-15-42n-28 n^{2}\big)B_{n,3}.
\end{multline*}

The same holds if $u\in I_{n,2}^{\prime \prime}$ and $v\in\left[\frac{7+26 n+28 n^{2}-u\left(6+26 n+28 n^{2}\right)}{3+10 n+7 n^{2}}, \frac{17+56 n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)}{7(1+n)(1+2 n)}\right]$.
Moreover, in both cases, we have

P(u,v)\cdot\mathbf{e}_1=14(1+n)(1+2 n)\left(17+56 n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2 n)\right)

\big(P(u,v)\big)^{2}=2 \left(17+56 n+56 n^{2}-u\left(15+56 n+56 n^{2}\right)-7v(1+n)(1+2n)\right)^{2} .

Thus, if $u\in I_{n,2}$, then

Now, we deal with $u\in I_{n,3}$.
If $u\in I_{n,3}^{\prime}$ and $v\in\left[0,\frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{59+112 n+56 n^{2}-u\left(57+112 n+56n^{2}\right)-7v(1+n)(3+2n)}{7(1+n)(3+2n)}\mathbf{e}_{1}+ \\
+\frac{39+91 n+49 n^{2}-u\left(48+98 n+49 n^{2}\right)}{7(1+n)(3+2n)}B_{n, 3}+\frac{u(1+n)(3+7n)-(1+n)(2+7n)}{7(1+n)(3+2n)}B_{n,4}
\end{multline*}

and $N(u,v)=0$.
The same holds if $u\in I_{n,3}^{\prime\prime}$ and $v\in\left[0,\frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}\right]$. Then

\begin{align*}
\big(P(u,v)\big)^{2}&=10-12u+2u^{2}-2v-v^{2},\\
\end{align*}

If $u\in I_{n,3}^\prime$ and $v \in\left[\frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}, \frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{59+112n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2n)}{7(1+n)(3+2n)}\mathbf{e}_{1}+\\
+\frac{\left(6+14n+7 n^{2}\right)\big(59+112n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2n)\big)}{7(1+n)(3+2n)}B_{n, 3}+\\
\end{multline*}

N(u,v)=\big(u\left(14+42 n+28 n^{2}\right)+v\left(6+14 n+7 n^{2}\right)-15-42n-28n^{2}\big)B_{n,3},

\begin{multline*}
\big(P(u, v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+\left(u\left(14+42 n+28 n^{2}\right)+v\left(6+14 n+7 n^{2}\right)-15-42n-28 n^{2}\right)^{2},
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=7(1+n)\big(13+53 n+70 n^{2}+28 n^{3}\big)-\\
-14u(1+n)(1+2n)\left(6+14 n+7 n^{2}\right)-7v(1+n)^2\left(5+14n+7n^{2}\right).
\end{multline*}

Similarly, if $u\in I_{n,3}^{\prime\prime}$ and $v \in\left[\frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}, \frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}\right]$, then

\begin{multline*}
\end{multline*}

N(u,v)=\big(u\left(30+58n+28n^{2}\right)+v\left(11+18n+7 n^{2}\right)-31-58n-28n^{2}\big)B_{n,4},

\begin{multline*}
\big(P(u, v)\big)^{2}=10-12 u+2 u^{2}-2v-v^{2}+\\
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=2388+8372 n+10983 n^{2}+6370 n^{3}+1372 n^{4}-\\
-14u(1+n)^{2}(11+7n)(15+14n)-v\left(846+2772 n+3346 n^{2}+1764 n^{3}+343 n^{4}\right).
\end{multline*}

If $u\in I_{n,3}^{\prime}$ and $v\in\left[\frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}, \frac{59+112 n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)}{7(1+n)(3+2 n)}\right]$, then

\begin{multline*}
P(u, v)=\frac{59+112 n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2n)}{7(1+n)(3+2n)}\mathbf{e}_1+\\
+\frac{\left(6+14 n+7 n^{2}\right)\big(59+112 n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2n)\big)}{7(1+n)(3+2n)}B_{n,3 }+\\
+\frac{(1+n)\left(11+7n\right)\big(59+112 n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2n)\big)}{7(1+n)(3+2n)}B_{n, 4}
\end{multline*}

\begin{multline*}
N(u,v)=\left(u\left(14+42 n+28 n^{2}\right)+v\left(6+14 n+7 n^{2}\right)-15-42n-28n^{2}\right)B_{n,3}+\\
+\left(u\left(30+58n+28n^{2}\right)+v\left(11+18n+7n^{2}\right)-31-58n-28n^{2}\right)B_{n, 4}.
\end{multline*}

The same holds if $u\in I_{u,3}^{\prime\prime}$ and $v\in\left[\frac{15+42 n+28 n^{2}-u\left(14+42 n+28 n^{2}\right)}{6+14 n+7 n^{2}}, \frac{59+112 n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)}{7(1+n)(3+2n)}\right]$.
In both cases, we have

P(u,v)\cdot\mathbf{e}_1=14(1+n)(3+2n)\left(59+112n+56 n^{2}-u\left(57+112 n+56 n^{2}\right)-7v(1+n)(3+2 n)\right).

\big(P(u,v)\big)^{2}=2\left(59+112n+56n^{2}-u\left(57+112n+56n^{2}\right)-7v(1+n)(3+2n)\right)^{2}.

Therefore, if  $u\in I_{n,3}$, then

Finally, we deal with $u\in I_{n,4}$.
If $u\in I_{n,4}^{\prime}$ and $v\in\left[0,\frac{31+58n+28n^{2}-u\left(30+58n+28n^{2}\right)}{11+18n+7n^{2}}\right]$, then

\begin{multline*}
P(u, v)=\frac{103+182 n+84 n^{2}-u\left(100+182 n+84 n^{2}\right)-v\left(36+56 n+21 n^{2}\right)}{36+56n+21 n^{2}}\mathbf{e}_{1}+\\
+\frac{17+32 n+14 n^{2}-u\left(20+34 n+14 n^{2}\right)}{36+56n+21n^{2}}B_{n,4}+\frac{u\left(48+98 n+49 n^{2}\right)-39-91n-49n^{2}}{36+56n+21 n^{2}}B_{n+1,1}
\end{multline*}

and $N(u,v)=0$. The same holds when $u\in I_{n,4}^{\prime\prime}$ and $v\in\left[0,\frac{44+70 n+28 n^{2}-u\left(43+70 n+28 n^{2}\right)}{15+21 n+7 n^{2}}\right]$.
In both cases, we compute

\begin{align*}
\big(P(u,v)\big)^{2}&=10-12u+2u^{2}-2 v-v^{2},\\
\end{align*}

If $u\in I_{n,4}^\prime$ and $v \in\left[\frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}, \frac{44+70 n+28 n^{2}-u\left(43+70 n+28 n^{2}\right)}{15+21 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u, v)=\frac{103+182n+84n^{2}-u(100+182n+84n^{2})-v(36+56n+21n^{2})}{36+56n+21n^{2}}\mathbf{e}_{1}+\\
\end{multline*}

N(u,v)=\left(u\left(30+58 n+28 n^{2}\right)+v\left(11+18 n+7 n^{2}\right)-31-58n-28n^{2}\right)B_{n,4},

\begin{multline*}
\big(P(u,v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+7\left(u\left(30+58 n+28 n^{2}\right)+v\left(11+18 n+7 n^{2}\right)-31-58n-28n^{2}\right)^{2},
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=2388+8372 n+10983 n^{2}+6370 n^{2}+1372 n^{4}-\\
-14u(1+n)^{2}(11+7n)(15+14 n)-\\
-v\left(846+2772 n+3346 n^{2}+1764 n^{3}+343 n^{4}\right).
\end{multline*}

If $u\in I_{n,4}^{\prime\prime}$ and $v \in\left[\frac{44+70 n+28 n^{2}-u\left(43+70 n+28 n^{2}\right)}{15+21 n+7 n^{2}}, \frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}}\right]$, then

\begin{multline*}
P(u,v)=\frac{103+182n+84n^{2}-u(100+182n+84n^{2})-v(36+56n+21n^{2})}{36+56 n+21n^{2}}\mathbf{e}_{1}+\\
+\frac{(15+21n+7n^{2})\big(103+182n+84n^{2}-u(100+182n+84n^{2})-v(36+56n+21n^{2})\big)}{36+56 n+21n^{2}}B_{n+1,1}+\\
+\frac{17+32n+14n^{2}-u\left(20+34n+14n^{2}\right)}{36+56 n+21n^{2}}B_{n,4}\quad\quad\quad\quad\quad\quad\quad
\end{multline*}

N(u,v)=\left(u\left(43+70 n+28 n^{2}\right)+v\left(15+21 n+7 n^{2}\right)-44-70n-28 n^{2}\right)B_{n+1,1}.

Moreover, in this case, we have

\begin{multline*}
\big(P(u,v)\big)^{2}=10-12 u+2 u^{2}-2 v-v^{2}+\\
+2\left(u\left(43+70 n+28 n^{2}\right)+v\left(15+21 n+7 n^{2}\right)-44-70n-28n^{2}\right)^{2}\quad\quad
\end{multline*}

\begin{multline*}
P(u,v)\cdot\mathbf{e}_1=1321+3948 n+4396 n^{2}+2156 n^{3}+392 n^{4}-\\
-2u\left(15+21 n+7 n^{2}\right)\left(43+70 n+28 n^{2}\right)-\\
-v\left(449+1260 n+1302 n^{2}+588 n^{3}+98 n^{4}\right).
\end{multline*}

If $u\in I_{n,4}^{\prime}$ and $v\in\left[\frac{44+70 n+28 n^{2}-u\left(43+70 n+28 n^{2}\right)}{15+21 n+7 n^{2}},\frac{103+182 n+84 n^{2}-u\left(100+182 n+84 n^{2}\right)}{36+56 n+21 n^{2}}\right]$, then

\begin{multline*}
+\frac{(15+21 n+7n^{2})\big(103+182n+84n^{2}-u(100+182n+84n^{2})-v(36+56n+21n^{2})\big)}{36+56n+21n^{2}}B_{n+1,1}
\end{multline*}

\begin{multline*}
N(u,v) =\big(u(30+58 n+28 n^{2})+v(11+18 n+7 n^{2})-31-58n-28n^{2}\big)B_{n,4} +\\
+\big(u(43+70 n+28 n^{2})+v(15+21 n+7 n^{2})-44-70n-28n^{2}\big)B_{n+1,1}.
\end{multline*}

The same holds when $u\in I_{n 4}^{\prime\prime}$ and $v \in\left[\frac{31+58 n+28 n^{2}-u\left(30+58 n+28 n^{2}\right)}{11+18 n+7 n^{2}},\frac{103+182 n+84 n^{2}-u\left(100+182 n+84 n^{2}\right)}{36+56 n+21 n^{2}}\right]$.
Moreover, in both cases, we have

\big(P(u,v)\big)^{2}=\big(103+182n+84n^{2}-u(100+182n+84 n^{2})-v(36+56 n+21 n^{2})\big)^{2}

Thus, if $u\in I_{n,4}$, then

Now, we are ready to compute $S(W_{\bullet,\bullet}^{T};\mathbf{e}_1)$.
Namely, for every $i\in\{1,2,3,4\}$, we set

On the other hand, integrating, we get

\left\{\aligned
&\frac{84365}{114688}\ \text{if $n=0$}, \\
&\frac{(8+28n+21n^{2})A_{n,1}}{448 n^{4}(1+n)(2+7n)^{4}(3+7n)^{4}(4+7n)^{4} (1+7n+7n^{2})}\ \text{if $n\geqslant 1$}, \\
\endaligned
\right.

\begin{multline*}
A_{n,1}=1536+109312 n+2935552 n^{2}+42681728 n^{3}+386407488 n^{4}+2335296292 n^{5}+\\
+9789648099 n^{6}+29038364761 n^{7}+61312905318 n^{8}+91454579804 n^{9}+\\
+94035837280 n^{10}+63317750608 n^{11}+25088413952 n^{12}+4427367168 n^{13}.\quad\quad
\end{multline*}

Similarly, we get

S_{n,2}=\frac{(1+2n)A_{n,2}}{4(1+n)(2+7n)^{4}(3+7n)^{4}(4+7n)^{4} (6+14n+7n^{2})},

\begin{multline*}
A_{n,2}=1618654+31459234 n+271069253 n^{2}+1362423916 n^{3}+4419070194 n^{4}+9654348284n^{5}+\\
+14368501182 n^{6}+14362052096 n^{7}+9209328422 n^{8}+3412762192n^9+553420896n^{10}
\end{multline*}

Likewise, we get

S_{n,3}=\frac{(3+2n)A_{n,3}}{4(1+n)(3+7n)^{4}(6+7n)^{4}(8+7n)^{4}(11+7n)(6+14 n+7 n^{2})},

\begin{multline*}
A_{n,3}=1167997914+15454923336 n+91492878645 n^{2}+319934133575 n^{3}+\\
+734395997090 n^{4}+1162203105378 n^{5}+1294197714054 n^{6}+1014406754242 n^{7}+\\
+548632346402 n^{8}+195059453722 n^{9}+41045383120 n^{10}+3873946272n^{11}.\quad\quad
\end{multline*}

Finally, we get

S_{n,4}=\frac{(36+56 n+21 n^{2})A_{n,4}}{448(1+n)^{4}(6+7n)^{4}(8+7n)^{4}(10+7n)^{4}(11+7n)(15+21n+7 n^{2})},

\begin{multline*}
A_{n,4}=365613573312+4021500121920 n+20341847967024 n^{2}+\\
+62650071283024 n^{3}+131072047236004 n^{4}+196698030664492 n^{5}+217823761840153 n^{6}+\\
+180219167765455 n^{7}+111395400841326 n^{8}+50802960251820 n^{9}+16615457209344 n^{10}+\\
+3690223711216 n^{11}+498816700928 n^{12}+30991570176 n^{13}.\quad\quad\quad\quad
\end{multline*}

Then, adding, we get

S\big(W_{\bullet,\bullet}^{T};\mathbf{e}_1\big)=\sum_{n=0}^{\infty}\Big(S_{n,1}+S_{n,2}+S_{n,3}+S_{n,4}\Big)\approx 0.976712233\ldots<1.

Finally, let us compute $S(W_{\bullet, \bullet,\bullet}^{T,\mathbf{e}_1};P)$.
For every $i\in\{1,2,3,4\}$, we set

\begin{align*}
\end{align*}

S\big(W_{\bullet, \bullet,\bullet}^{T,\mathbf{e}_1};P\big)=
\sum_{n=0}^{\infty}\sum_{i=1}^{4}\Big(M_{n,i}^\prime+M_{n,i}^{\prime\prime}\Big)+\frac{3}{7}\int_{0}^1\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_P\Big(N(u,v)\big|_{\mathbf{e}_1}\Big)dvdu.

On the other hand, integrating, we get

\left\{\aligned
&\frac{1403}{22268}\ \text{if $n=0$}, \\
&\frac{(1+n)A_{n,1}^\prime}{448n^{4}(1+3n)^{4}(3+7n)^{4}(1+7n+7 n^{2})}\ \text{if $n\geqslant 1$}, \\
\endaligned
\right.

\begin{multline*}
A_{n,1}^\prime=1+81 n+2535 n^{2}+37209 n^{3}+301046 n^{4}+1459736 n^{5}+\\
+4420190 n^{6}+8425410 n^{7}+9821448 n^{8}+6392736 n^{9}+1778112 n^{10}.\quad\quad\quad\quad
\end{multline*}

Similarly, we get

M_{n, 1}^{\prime\prime}=\frac{(1+7 n+7 n^{2})A_{n,1}^{\prime\prime}}{28(1+n)(1+3 n)^{4}(2+7 n)^{4}(3+7 n)^{4}(4+7 n)^{4}},

\begin{multline*}
A_{n,1}^{\prime\prime}=480574+12906866 n+157271760 n^{2}+1149521334 n^{3}+5612285145 n^{4}+\\
+19278934535 n^{5}+47770884833 n^{6}+86016481159 n^{7} +111679016743 n^{8}+\\
+101939513907 n^{9}+62077730148 n^{10}+22635902898 n^{11}+3735591048 n^{12}.
\end{multline*}

Likewise, we have

M_{n,2}^{\prime}=\frac{(6+14n+7n^{2})A_{n,2}^{\prime}}{224(1+n)(1+2 n)^{3}(2+7 n)^{4}(3+7 n)^{4}(4+7 n)^{4}},

M_{n,2}^{\prime\prime}=\frac{11780+111142 n+430951 n^{2}+875637 n^{3}+978656 n^{4}+566832 n^{5}+131712 n^{6}}{224(1+2 n)^{3}(3+7 n)^{4}( 6+14 n+7 n^{2})}.

\begin{multline*}
A_{n,2}^{\prime}=1561176+35176776 n+356105548 n^{2}+2137950448 n^{3}+\\
+8458603286 n^{4}+23158717414 n^{5}+44778314889 n^{6}+61151030584 n^{7}+\\
+57807289939 n^{8}+36026947376 n^{9}+13321631568 n^{10}+2213683584 n^{11}.
\end{multline*}

Similarly, we have

M_{n,3}^{\prime}=\frac{(11+7n)A_{n,3}^{\prime}}{224(1+n)^{3}(3+7 n)^{4}(13+14 n)^{4} (6+14 n+7 n^{2})},

\begin{multline*}
A_{n,3}^{\prime}=13726028+164541190 n+859036123 n^{2}+2564002455 n^{3}+4823323519 n^{4}+\\
+5933644367 n^{5}+4776917782 n^{6}+2428774768 n^{7}+708314208 n^{8}+90354432 n^{9}.
\end{multline*}

Likewise, we have

M_{n,3}^{\prime\prime}=\frac{(6+14 n+7 n^{2})A_{n,3}^{\prime\prime}}{224(1+n)^{3}(6+7 n)^{4}(8+7 n)^{4}(11+7 n)(13+14n)^{4}},

\begin{multline*}
A_{n,3}^{\prime\prime}=67760261208+703706084640 n+3313300067388 n^{2}+9335574166156 n^{3}+\\
+17489294547578 n^{4}+22873117200584 n^{5}+21308562209725 n^{6}+14139587568253 n^{7}+\\
+6548997703738 n^{8}+2016283621072 n^{9}+371345421216 n^{10}+30991570176 n^{11}.
\end{multline*}

Similarly, we see that

M_{n,4}^{\prime}=\frac{(15+21n+7n^{2})A_{n,4}^{\prime}}{28(1+n)^{4}(6+7 n)^{4}(8+7 n)^{4}(11+7 n)(23+21 n)^{4}},

\begin{multline*}
A_{n,4}^{\prime}=88135013250+967134809574 n+4853884596732 n^{2}+\\
+14732868828434 n^{3}+30120687035243 n^{4}+43697011451345 n^{5}+46124583653603 n^{6}+\\
+35692827118809 n^{7}+20096052100397 n^{8}+8028312817917 n^{9}+\\
+2160120347280 n^{10}+351456857766 n^{11}+26149137336 n^{12}.\quad\quad\quad\quad
\end{multline*}

Finally, we have

M_{n,4}^{\prime\prime}=\frac{(11+7n)A_{n,4}^{\prime\prime}}{448(1+n)^{4}(10+7 n)^{4}(23+21 n)^{4} (15+21 n+7 n^{2})},

\begin{multline*}
A_{n,4}^{\prime\prime}=7582266167+59702225967 n+210973884925 n^{2}+440580768679 n^{3}+\\
+602090743422 n^{4}+562572998512 n^{5}+363945674554 n^{6}+160955181870 n^{7}+\\
+46566357768 n^{8}+7957643904 n^{9}+609892416 n^{10}.\quad\quad\quad\quad\quad\quad\quad
\end{multline*}

Now, adding terms together, we see that

\begin{equation}
\label{equation:2-7-final}
S\big(W_{\bullet, \bullet,\bullet}^{T,\mathbf{e}_1};P\big)\leqslant 0.974+\frac{3}{7}\int_{0}^1\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_P\Big(N(u,v)\big|_{\mathbf{e}_1}\Big)dvdu.
\end{equation}

Now, for every $i\in\{1,2,3,4\}$ and any irreducible component $\ell$ of the curve $B_{n,i}$, we let

\begin{multline*}
\quad\quad\quad\quad\quad F_{n,i}=\frac{3}{7}\int_{0}^1\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_\ell\big(N(u,v)\big)\big(\ell\cdot\mathbf{e}_1\big)dvdu=\\
\quad=\frac{3}{7}\int_{I_{n,i-1}}\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_\ell\big(N(u,v)\big)\big(\ell\cdot\mathbf{e}_1\big)dvdu+\\
+\frac{3}{7}\int_{I_{n,i}}\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_\ell\big(N(u,v)\big)\big(\ell\cdot\mathbf{e}_1\big)dvdu+\\
+\frac{3}{7}\int_{I_{n,i+1}}\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_\ell\big(N(u,v)\big)\big(\ell\cdot\mathbf{e}_1\big)dvdu,\quad\quad\quad\quad\quad\quad
\end{multline*}

where  $I_{0,0}=\varnothing$ and $I_{n,5}=I_{n+1,1}$. Since irreducible components of  $B_{n,i}$ are disjoint, we get

\frac{3}{7}\int_{0}^1\int_0^{t(u)}\big(P(u,v)\cdot \mathbf{e}_1\big)\mathrm{ord}_P\Big(N(u,v)\big|_{\mathbf{e}_1}\Big)dvdu\leqslant \sum_{n=0}^{\infty}\sum_{i=1}^{4}F_{n,i}.

On the other hand, each $F_{n,i}$ is not difficult to compute.
For instance, we have

\begin{multline*}
+\frac{3}{7}\int_{0}^{\frac{1}{3}}\int_{\frac{7-6 u}{3}}^{\frac{19-16u}{8}}8(u+v-2)(19-16u-8v)dvdu+\\
+\frac{3}{7}\int_{\frac{1}{3}}^{\frac{3}{8}}\int_{2-u}^{\frac{19-16 u}{8}}8(u+v-2)(19-16u-8v)dvdu=\frac{281}{32256}.
\end{multline*}

Similarly, we see that

F_{n,1}=\frac{3(1+7 n+7 n^{2} )^{2}}{2 n^{2}(1+3 n)(-1+7 n)(1+7 n)(2+7 n)(3+7 n)^{2}(4+7 n)(2+21 n)}

for $n\geqslant 1$. Likewise, we get

\left\{\aligned
&\frac{5}{3584}\ \text{if $n=0$}, \\
&\frac{1+n}{112 n(1+2 n)(1+3 n)(2+7 n)(3+7 n)^{2}(4+7 n)}\ \text{if $n\geqslant 1$}. \\
\endaligned
\right.

Likewise, for every $n\geqslant 0$, we have

F_{n,3}=\frac{15 (6+14n+7n^{2})^{2}}{4(1+n)^{2}(1+2 n)(2+7 n)(3+7 n)^{2}(4+7 n)(6+7 n)(8+7 n)(13+14 n)}

F_{n,4}=\frac{(11+7 n)^{2}}{112(1+n)^{2}(3+7 n)(6+7 n)(8+7 n)(10+7 n)(13+14 n)(23+21 n)}.

Now, one can easily check that the total sum of all $F_{n,1}$, $F_{n,2}$, $F_{n,3}$, $F_{n,4}$ is at most $0.014$.
This and (<ref>) give $S(W_{\bullet, \bullet,\bullet}^{T,\mathbf{e}_1};P)\leqslant 0.974+0.014=0.988$.
Using (<ref>), we get $\delta_P(X)>1$.

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