Page:Mémoires de l’Académie des sciences, Tome 10.djvu/818

d’où l’on tire

${\begin{aligned}\mathrm {A} &={\frac {1}{8\pi ^{3}}}\left(\alpha f(x',y',z')+\beta f'(x',y',z')+\gamma f''(x',y',z')\right),\\\mathrm {B} &={\frac {1}{8\pi ^{3}}}f'(x',y',z')-{\frac {\beta }{8\pi ^{3}}}\left(\alpha f(x',y',z')+\beta f'(x',y',z')+\gamma f''(x',y',z')\right),\\\mathrm {C} &={\frac {1}{8\pi ^{3}}}f''(x',y',z')-{\frac {\gamma }{8\pi ^{3}}}\left(\alpha f(x',y',z')+\beta f'(x',y',z')+\gamma f''(x',y',z')\right).\end{aligned}}$

Les valeurs de $\mathrm {A',B',C} ',$ se déduiront de celles de $\mathrm {A,B,C} ,$ en y remplaçant les fonctions $f,f',f'',$ par $\operatorname {F,F',F''} .$ Ces six quantités étant ainsi déterminées, si l’on fait, pour abréger,

$\mathrm {T} ={\frac {1}{8\pi ^{3}}}\left(\alpha f(x',y',z')+\beta f'(x',y',z')+\gamma f''(x',y',z')\right)(\cos .\rho at-\cos .\rho bt)$

$+{\frac {1}{8\pi ^{3}}}\left(\alpha \operatorname {F} (x',y',z')+\beta \operatorname {F} '(x',y',z')+\gamma \operatorname {F} ''(x',y',z')\right)\left({\frac {\sin .\rho at}{\rho a}}-{\frac {\sin .\rho bt}{\rho b}}\right),$

les formules (5) deviendront

$\left.{\begin{aligned}u=&\iiint \!\!\!\iiint \mathrm {T} \cos .\rho \delta .\alpha \rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz'\\&+\iiint \!\!\!\iiint {\bigg (}f(x',y',z')\cos .\rho bt{\bigg .}\\&+\left.\operatorname {F} (x',y',z'){\frac {\sin .\rho bt}{\rho b}}\right)\cos .\rho \delta .\rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz',\\v=&\iiint \!\!\!\iiint \mathrm {T} \cos .\rho \delta .\beta \rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz'\\&+\iiint \!\!\!\iiint {\bigg (}f'(x',y',z')\cos .\rho bt{\bigg .}\\&+\left.\operatorname {F} '(x',y',z'){\frac {\sin .\rho bt}{\rho b}}\right)\cos .\rho \delta .\rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz',\\w=&\iiint \!\!\!\iiint \mathrm {T} \cos .\rho \delta .\gamma \rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz'\\&+\iiint \!\!\!\iiint {\bigg (}f''(x',y',z')\cos .\rho bt{\bigg .}\\&+\left.\operatorname {F} ''(x',y',z'){\frac {\sin .\rho bt}{\rho b}}\right)\cos .\rho \delta .\rho ^{2}\sin .\theta \,d\rho \,d\theta \,d\omega \,dx'dy'dz'.\end{aligned}}\right\}$

(6)