Page:Mémoires de l’Académie des sciences, Tome 8.djvu/636

les équations (1) deviendront :

${\begin{aligned}\mathrm {P} &=\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }\left[(g+g')\sum {\frac {r^{3}}{\alpha ^{5}}}fr+(gg'+hh'+ll')g\sum {\frac {r^{5}}{\alpha ^{5}}}\,{\frac {d.{\frac {1}{r}}fr}{dr}}\right]\Delta ,\\\mathrm {Q} &=\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }\left[(h+h')\sum {\frac {r^{3}}{\alpha ^{5}}}fr+(gg'+hh'+ll')h\sum {\frac {r^{5}}{\alpha ^{5}}}\,{\frac {d.{\frac {1}{r}}fr}{dr}}\right]\Delta ,\\\mathrm {R} &=\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }\left[(l+l')\sum {\frac {r^{3}}{\alpha ^{5}}}fr+(gg'+hh'+ll')l\sum {\frac {r^{5}}{\alpha ^{5}}}\,{\frac {d.{\frac {1}{r}}fr}{dr}}\right]\Delta ,\end{aligned}}$

où l’on a mis, pour abréger, $\Delta$ à la place de $\cos .\beta \sin .\beta \,d\beta \,d\gamma .$

On aura immédiatement

${\begin{aligned}\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }(g+g')\Delta &=-{\frac {2\pi }{3}}\left(c+c{\frac {du}{dx}}+c'{\frac {du}{dy}}+c''{\frac {du}{dz}}\right),\\\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }(h+h')\Delta &=-{\frac {2\pi }{3}}\left(c'+c{\frac {dv}{dx}}+c'{\frac {dv}{dy}}+c''{\frac {dv}{dz}}\right),\\\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }(l+l')\ \Delta &=-{\frac {2\pi }{3}}\left(c''+c{\frac {dw}{dx}}+c'{\frac {dw}{dy}}+c''{\frac {dw}{dz}}\right).\end{aligned}}$

On aura aussi, sans difficulté,

${\begin{aligned}\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }g^{3}\Delta &=-{\frac {2\pi c}{5}}\left(a^{2}+b^{2}+c^{2}\right),\\\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }g^{2}h\Delta &=-{\frac {2\pi }{15}}\left(3c^{2}c'+2aa'c+2bb'c+a^{2}c'+b^{2}c'\right),\\\int _{0}^{{\frac {1}{2}}\pi }\int _{0}^{2\pi }ghl\Delta &=-{\frac {2\pi }{15}}(3cc'c''+aa'c''+aa''c'+a'a''c+bb'c''+bb''c'+b'b''c)\,;\end{aligned}}$