Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 12.djvu/187

donc

${\begin{aligned}&\operatorname {\mathrm {S} } \alpha r^{2}drd\mu d\varpi \left(y'{\frac {\partial \mathrm {U} }{\partial z'}}-z'{\frac {\partial \mathrm {U} }{\partial y'}}\right),\\&\operatorname {\mathrm {S} } \alpha r^{2}drd\mu d\varpi \left(z'{\frac {\partial \mathrm {U} }{\partial x'}}-x'{\frac {\partial \mathrm {U} }{\partial z'}}\right),\\&\operatorname {\mathrm {S} } \alpha r^{2}drd\mu d\varpi \left(x'{\frac {\partial \mathrm {U} }{\partial y'}}-y'{\frac {\partial \mathrm {U} }{\partial x'}}\right).\end{aligned}}$

Pour intégrer ces fonctions relativement à $r,$ nous observerons que, la profondeur de la mer étant supposée très petite, on peut supposer ici

$\operatorname {S} r^{2}dr=\gamma .$

Si, de plus, on change les différences partielles ${\frac {\partial \mathrm {U} }{\partial x'}},{\frac {\partial \mathrm {U} }{\partial y'}}$ et ${\frac {\partial \mathrm {U} }{\partial z'}}$ en d’autres, relatives aux variables $r,\varpi$ et $\mu$ les fonctions précédentes deviendront, en les prenant avec un signe contraire,

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

Si l’on réunit ces valeurs aux expressions partielles de ${\frac {d\mathrm {N} }{dt}},{\frac {d\mathrm {N} '}{dt}},{\frac {d\mathrm {N} ''}{dt}},$ trouvées ci-dessus, on aura, pour les expressions entières de ces quantités relatives à l’attraction et à la pression de l’océan,

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