Page:Mémoires de l’Académie des sciences, Tome 8.djvu/637

et comme on a

$a^{2}+b^{2}+c^{2}=1,\qquad aa'+bb'+cc'=0,$ etc.,

ces valeurs se réduiront à

$\int \int g^{3}\Delta =-{\frac {2\pi c}{5}},\quad \int \int g^{2}h\Delta =-{\frac {2\pi c'}{15}},\quad \int \int ghl\Delta =0.$

On trouvera de même

${\begin{array}{lll}\int \int h^{3}\Delta =-{\frac {2\pi c'}{5}},&\int \int l^{3}\Delta =-{\frac {2\pi c''}{5}},\\\int \int gh^{2}\Delta =-{\frac {2\pi c}{15}},&\int \int gl^{2}\Delta =-{\frac {2\pi c}{15}},&\int \int g^{2}l\Delta =-{\frac {2\pi c''}{15}},\\\int \int h^{2}l\Delta =-{\frac {2\pi c''}{15}},&\int \int hl^{2}\Delta =-{\frac {2\pi c'}{15}}.\end{array}}$

Au moyen de ces différents résultats, et en faisant, pour abréger,

${\frac {2\pi }{3}}\sum {\frac {r^{3}}{\alpha ^{5}}}fr=\mathrm {K} ,\qquad {\frac {2\pi }{15}}\sum {\frac {r^{5}}{\alpha ^{5}}}{\frac {d.{\frac {1}{r}}fr}{dr}}=k,$

nous aurons définitivement

${\begin{aligned}\mathrm {P} &=-\mathrm {K} \left(c+{\frac {du}{dx}}c+{\frac {du}{dy}}c'+{\frac {du}{dz}}c''\right)\\&=-k\left(3{\frac {du}{dx}}c+{\frac {du}{dy}}c'+{\frac {du}{dz}}c''+{\frac {dv}{dx}}c'+{\frac {dv}{dy}}c+{\frac {dw}{dx}}c''+{\frac {dw}{dz}}c\right),\\\mathrm {Q} &=-\mathrm {K} \left(c'+{\frac {dv}{dx}}c+{\frac {dv}{dy}}c'+{\frac {dv}{dz}}c''\right)\\&=-k\left(3{\frac {dv}{dy}}c'+{\frac {dv}{dx}}c+{\frac {dv}{dz}}c''+{\frac {du}{dx}}c'+{\frac {du}{dy}}c+{\frac {dw}{dy}}c''+{\frac {dw}{dz}}c'\right),\\\mathrm {R} &=-\mathrm {K} \left(c''+{\frac {dw}{dx}}c+{\frac {dw}{dy}}c'+{\frac {dw}{dz}}c''\right)\\&=-k\left(3{\frac {dw}{dz}}c''+{\frac {dw}{dx}}c+{\frac {dw}{dy}}c'+{\frac {du}{dx}}c''+{\frac {du}{dz}}c+{\frac {dv}{dy}}c''+{\frac {dv}{dz}}c'\right),\\\end{aligned}}$