Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/202

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La valeur de $\mathrm {D}$ est facile à déterminer lorsqu’on connaît le rayon $1+\alpha y$ du sphéroïde, car on a, par l’article I,

{\begin{aligned}\mathrm {A} =&\int _{0}^{\pi }\int _{0}^{\pi }\left[2\sin ^{3}p\sin ^{2}q(1+\alpha y)+\alpha (y'-y)\sin p\right]dpdq\\=&{\frac {4}{3}}\pi -{\frac {2}{3}}\alpha \pi y+\alpha \int _{0}^{\pi }\int _{0}^{\pi }y'\sin pdpdq,\end{aligned}}

ce qui donne

\mathrm {D} =2\alpha \int _{0}^{\pi }\int _{0}^{\pi }y'\sin pdpdq.

XXIII.

Si l’on suppose $n=0$ et $\gamma =1,$ les équations (6), (7) et (9) de l’article précédent, deviendront

{\begin{aligned}y\sin \theta =&-l{\frac {\partial .u\sin \theta }{\partial \theta }}-l{\frac {\partial v}{\partial \varpi }}\sin \theta ,\\{\frac {d^{2}u}{dt^{2}}}=&-g{\frac {\partial y}{\partial \theta }}+{\frac {\partial \mathrm {D} }{\partial \theta }}\Delta +{\frac {\partial \mathrm {R} }{\partial \theta }},\\{\frac {d^{2}v}{dt^{2}}}\sin ^{2}\theta =&-g{\frac {\partial y}{\partial \varpi }}+{\frac {\partial \mathrm {D} }{\partial \varpi }}\Delta +{\frac {\partial \mathrm {R} }{\partial \varpi }}.\end{aligned}}

La seconde de ces équations donne

{\begin{aligned}-l{\frac {d^{2}}{dt^{2}}}{\frac {\partial .u\sin \theta }{\partial \theta }}=&lg{\frac {\partial ^{2}y}{\partial \theta ^{2}}}\sin \theta +lg{\frac {\partial y}{\partial \theta }}\cos \theta \\&-l{\frac {\partial ^{2}\mathrm {D} }{\partial \theta ^{2}}}\Delta \sin \theta -l{\frac {\partial \mathrm {D} }{\partial \theta }}\Delta \cos \theta \\&-l{\frac {\partial ^{2}\mathrm {R} }{\partial \theta ^{2}}}\sin \theta -l{\frac {\partial \mathrm {R} }{\partial \theta }}\cos \theta .\end{aligned}}

La troisième donne

-l{\frac {d^{2}}{dt^{2}}}{\frac {\partial v}{\partial \varpi }}\sin \theta ={\frac {lg}{\sin \theta }}{\frac {\partial ^{2}y}{\partial \varpi ^{2}}}-{\frac {l}{\sin \theta }}{\frac {\partial ^{2}\mathrm {D} }{\partial \varpi ^{2}}}\Delta -{\frac {l}{\sin \theta }}{\frac {\partial ^{2}\mathrm {R} }{\partial \varpi ^{2}}}.