Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 4.djvu/92

On aura donc

$an\operatorname {d} {\rm {R}}=$

$-d(e\cos \varpi )\left\{{\begin{aligned}&\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]e\cos \varpi \\&-{\begin{array}{|c|}\hline 0,\ 1\\\hline \end{array}}e'\cos \varpi '-{\begin{array}{|c|}\hline 0,\ 2\\\hline \end{array}}e''\cos \varpi ''-{\begin{array}{|c|}\hline 0,\ 3\\\hline \end{array}}e'''\cos \varpi '''\end{aligned}}\right\}$

$-d(e\sin \varpi )\left\{{\begin{aligned}&\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]e\sin \varpi \\&-{\begin{array}{|c|}\hline 0,\ 1\\\hline \end{array}}e'\sin \varpi '-{\begin{array}{|c|}\hline 0,\ 2\\\hline \end{array}}e''\sin \varpi ''-{\begin{array}{|c|}\hline 0,\ 3\\\hline \end{array}}e'''\sin \varpi '''\end{aligned}}\right\}$

$-d(\gamma _{1}\cos {\text{⅂}}_{1})\left\{{\begin{aligned}&\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]\gamma _{1}\cos {\text{⅂}}_{1}\\&+(0)\theta \cos \Psi -{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}\gamma \cos {\text{⅂}}\\&-(0,\ 1)\gamma '_{1}\cos {\text{⅂}}'_{1}-(0,\ 2)\gamma ''_{1}\cos {\text{⅂}}''_{1}-(0,\ 3)\gamma '''_{1}\cos {\text{⅂}}'''_{1}\end{aligned}}\right\}$

$+d(\gamma _{1}\sin {\text{⅂}}_{1})\left\{{\begin{aligned}&\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]\gamma _{1}\sin {\text{⅂}}_{1}\\&-(0)\theta \sin \Psi -{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}\gamma \sin {\text{⅂}}\\&-(0,\ 1)\gamma '_{1}\sin {\text{⅂}}'_{1}-(0,\ 2)\gamma ''_{1}\sin {\text{⅂}}''_{1}-(0,\ 3)\gamma '''_{1}\sin {\text{⅂}}'''_{1}\end{aligned}}\right\}$

On a, par le n^o 6,

${\begin{aligned}e\sin \varpi =&-h\sin(gt+\Gamma )-\ldots ,\\e\cos \varpi =&-h\cos(gt+\Gamma )-\ldots \,;\end{aligned}}$

les équations entre $h,h',\ldots$ du numéro cité donneront ainsi les suivantes

${\begin{aligned}{\frac {d(e\sin \varpi )}{dt}}&=\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]e\cos \varpi \\&-{\begin{array}{|c|}\hline 0,\ 1\\\hline \end{array}}e'\cos \varpi '-{\begin{array}{|c|}\hline 0,\ 2\\\hline \end{array}}e''\cos \varpi ''-{\begin{array}{|c|}\hline 0,\ 3\\\hline \end{array}}e'''\cos \varpi ''',\\\\{\frac {d(e\cos \varpi )}{dt}}&=\left[(0)+{\begin{array}{|c|}\hline \ 0\ \\\hline \end{array}}+(0,\ 1)+(0,\ 2)+(0,\ 3)\right]e\sin \varpi \\&-{\begin{array}{|c|}\hline 0,\ 1\\\hline \end{array}}e'\sin \varpi '-{\begin{array}{|c|}\hline 0,\ 2\\\hline \end{array}}e''\sin \varpi ''-{\begin{array}{|c|}\hline 0,\ 3\\\hline \end{array}}e'''\sin \varpi '''.\end{aligned}}$

On a ensuite

$s=\gamma _{1}\sin(v-{\text{⅂}}_{1}).$

En comparant cette équation à celle-ci,

$s=l\sin(v+pt+\Lambda )+l_{1}\sin(v+p_{1}t+\Lambda _{1})+\ldots ,$

on a

${\begin{aligned}\gamma _{1}\sin {\text{⅂}}_{1})=&-l\sin(pt+\Lambda )-\ldots ,\\\gamma _{1}\cos {\text{⅂}}_{1})=&\quad l\cos(pt+\Lambda )+\ldots .\end{aligned}}$