Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/120

dantes, elle donnera les suivantes

(7)

$\left\{{\begin{aligned}{\frac {d^{2}u}{dt^{2}}}&-2n{\frac {dv}{dt}}\sin \theta \cos \theta \\=&-g{\frac {\partial y}{\partial \theta }}+\mathrm {B} \Delta \\+&\mathrm {K} \left\{\sin 2\theta \left[{\frac {1}{2}}\sin ^{2}\nu -\cos ^{2}\nu +{\frac {1}{2}}\sin ^{2}\nu \cos(2\varphi -2nt-2\varpi )\right]\right.\\&\qquad \qquad \qquad \qquad \left.+2\cos 2\theta \sin \nu \cos \nu \cos(\varphi -nt-\varpi ){\begin{aligned}\\\\\end{aligned}}\right\}\end{aligned}}\right.$

ou, en substituant au lieu de $y$ sa valeur,

(8)

$\left\{{\begin{aligned}{\frac {d^{2}u}{dt^{2}}}&-2n{\frac {dv}{dt}}\sin \theta \cos \theta \\=&lg\gamma {\frac {\partial ^{2}u}{\partial \theta ^{2}}}+2lg{\frac {\partial \gamma }{\partial \theta }}{\frac {\partial u}{\partial \theta }}+lgu{\frac {\partial ^{2}\gamma }{\partial \theta ^{2}}}+{\frac {\partial ^{2}v}{\partial \varpi \partial \theta }}\\+&lg{\frac {\partial v}{\partial \varpi }}{\frac {\partial \gamma }{\partial \theta }}+lg\gamma {\frac {\partial .u{\cfrac {\cos \theta }{\sin \theta }}}{\partial \theta }}+lg{\frac {\partial \gamma }{\partial \theta }}{\frac {u\cos \theta }{\sin \theta }}+\mathrm {B} \Delta \\+\mathrm {K} &\left\{{\begin{aligned}\sin &2\theta \left[{\frac {1}{2}}\sin ^{2}\nu -\cos ^{2}\nu +{\frac {1}{2}}\sin ^{2}\nu \cos(2\varphi -2nt-2\varpi )\right]\\&+2\cos 2\theta \sin \nu \cos \nu \cos(\varphi -nt-\varpi )\end{aligned}}\right\},\end{aligned}}\right.$

(9)

$\left\{{\begin{aligned}{\frac {d^{2}v}{dt^{2}}}\sin ^{2}\theta +2n{\frac {du}{dt}}\sin \theta \cos \theta =&-g{\frac {\partial y}{\partial \theta }}+\mathrm {C} \Delta \sin \theta \\+\mathrm {K} &\sin \nu \cos \nu \sin 2\theta \sin(\varphi -nt-\varpi )\\+\mathrm {K} &\sin ^{2}\nu \sin ^{2}\theta \sin(2\varphi -2nt-2\varpi )\,;\end{aligned}}\right.$

ou, en substituant au lieu de $y$ sa valeur,

(10)

$\left\{{\begin{aligned}{\frac {d^{2}v}{dt^{2}}}&\sin ^{2}\theta +2n{\frac {du}{dt}}\sin \theta \cos \theta \\=&lg\gamma {\frac {\partial ^{2}v}{\partial \varpi ^{2}}}+lg\gamma {\frac {\partial ^{2}u}{\partial \varpi \partial \theta }}+lg\gamma {\frac {\cos \theta }{\sin \theta }}{\frac {\partial u}{\partial \varpi }}+lg{\frac {\partial \gamma }{\partial \theta }}{\frac {\partial u}{\partial \varpi }}+\mathrm {C} \Delta \sin \theta \\+&\mathrm {K} \sin \nu \cos \nu \sin 2\theta \sin(\varphi -nt-\varpi )\\+&\mathrm {K} \sin ^{2}\nu \sin ^{2}\theta \sin(2\varphi -2nt-2\varpi ).\end{aligned}}\right.$

Il faut ensuite satisfaire pour tous les points du fluide aux deux équations

(4)

${\frac {\partial \left({\cfrac {du}{dt}}-2nv\sin \theta \cos \theta \right)}{\partial \varpi }}={\frac {\partial \left(\sin ^{2}\theta {\cfrac {dv}{dt}}+2nu\sin \theta \cos \theta \right)}{\partial \theta }},$