Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 12.djvu/195

on aura

$\operatorname {S} \alpha \gamma d\mu d\varpi \left[{\frac {\partial ^{2}u}{\partial t^{2}}}\sin(\varphi +\varpi )+2n\mu ^{2}{\frac {\partial u}{\partial t}}\cos(\varphi +\varpi )\right.$

$\left.+{\frac {\partial ^{2}v}{\partial t^{2}}}\mu {\sqrt {1-\mu ^{2}}}\cos(\varphi +\varpi )-2n{\frac {\partial v}{\partial t}}\mu {\sqrt {1-\mu ^{2}}}\sin(\varphi -\varpi )\right]$

${\begin{aligned}=\quad \ &\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ \sin(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \mu }}\ -{\frac {\partial \mathrm {U} }{\partial \mu }}-{\frac {\partial f}{\partial \mu }}\right)\\-&\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\cos(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial \mathrm {U} }{\partial \varpi }}-{\frac {\partial f}{\partial \varpi }}\right).\end{aligned}}$

Si l’on substitue pour $\gamma u$ l’ensemble des termes relatifs à l’angle $it,$ et si l’on observe qu’ici nous ne considérons que le cas dans lequel diffère très peu de $n,$ le premier membre de cette équation deviendra

$-\alpha n^{2}\pi \operatorname {S} d\mu \cos(it+\varepsilon -\varphi )\left[\left(1-2\mu ^{2}\right)\mathrm {M} '-\mu {\sqrt {1-\mu ^{2}}}\mathrm {O} '\right]\,;$

on aura donc, en n’ayant égard qu’aux termes dans lesquels $i$ est à très peu près égal à $n,$

${\begin{aligned}\operatorname {S} \alpha n^{2}&yd\mu d\varpi \mu {\sqrt {1-\mu ^{2}}}\sin(\varphi +\varpi )\\=&\quad \ \operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\sin(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \mu }}\ \ -{\frac {\partial \mathrm {U} }{\partial \mu }}-{\frac {\partial f}{\partial \mu }}\right)\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\cos(\varphi +\varpi )\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial \mathrm {U} }{\partial \varpi }}-{\frac {\partial f}{\partial \varpi }}\right),\end{aligned}}$

d’où l’on tire

${\begin{aligned}{\frac {d\mathrm {N} ''}{dt}}\sin \varphi -{\frac {d\mathrm {N} '}{dt}}\cos \varphi =&\quad \operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}{\frac {\partial f}{\partial \varpi }}\cos(\varphi +\varpi )\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ {\frac {\partial f}{\partial \mu }}\sin(\varphi +\varpi ).\end{aligned}}$

On trouvera de la même manière

${\begin{aligned}{\frac {d\mathrm {N} ''}{dt}}\cos \varphi +{\frac {d\mathrm {N} '}{dt}}\sin \varphi =&-\operatorname {S} \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}{\frac {\partial f}{\partial \varpi }}\sin(\varphi +\varpi )\\&-\operatorname {S} \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\ \ {\frac {\partial f}{\partial \mu }}\cos(\varphi +\varpi ).\end{aligned}}$