![{\displaystyle {\begin{aligned}{\frac {d^{2}\varpi }{dx^{2}}}={\frac {d^{2}\varpi }{d\rho ^{2}}}-2{\frac {d^{2}\varpi }{d\rho d\theta }}&{\frac {\sin .\theta \cos .\theta }{\rho }}+{\frac {d^{2}\varpi }{d\theta ^{2}}}{\frac {\sin .^{2}\theta }{\rho ^{2}}}\\&+2{\frac {d\varpi }{d\theta }}{\frac {\sin .\theta \cos .\theta }{\rho }}+{\frac {d\varpi }{d\rho }}{\frac {\sin .^{2}\theta }{\rho }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6659f5b248dc9d6723fe3e9582c224d8762afb37)
Par des combinaisons analogues, on obtiendra les valeurs de
et
en différences partielles relatives à
En les ajoutant ensuite à la valeur de
on trouve, toutes réductions faites,
![{\displaystyle {\frac {d^{2}\varpi }{dx^{2}}}+{\frac {d^{2}\varpi }{dy^{2}}}+{\frac {d^{2}\varpi }{dz^{2}}}={\frac {1}{\rho }}{\frac {d^{2}.\rho \varpi }{d\rho ^{2}}}+{\frac {1}{\rho ^{2}\sin .\theta }}{\frac {d.{\frac {d\varpi }{d\theta }}\sin .\theta }{d\theta }}+{\frac {1}{\rho ^{2}\sin .^{2}\theta }}{\frac {d^{2}\varpi }{d\omega ^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c06afdad8e7486685d6c77ac512ccda5f043f3)
On conclut delà
![{\displaystyle \int _{0}^{at}\int _{0}^{\pi }\int _{0}^{2\pi }\left({\frac {d^{2}\varpi }{dx^{2}}}+{\frac {d^{2}\varpi }{dy^{2}}}+{\frac {d^{2}\varpi }{dz^{2}}}\right)\rho \sin .\theta \,d\rho \,d\theta \,d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f258900c3cb58c994a339f0cd59845815343348c)
![{\displaystyle =\int _{0}^{\pi }\int _{0}^{2\pi }\left(\int _{0}^{at}{\frac {d^{2}.\rho \varpi }{d\rho ^{2}}}d\rho \right)\sin .\theta \,d\theta \,d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fa04cfe80628eec56a36f9e4000cb4d9294a3d)
![{\displaystyle +\int _{0}^{at}\int _{0}^{2\pi }\left(\int _{0}^{\pi }{\frac {d.{\frac {d\varpi }{d\theta }}\sin .\theta }{d\theta }}d\theta \right){\frac {d\rho \,d\omega }{\rho }}+\int _{0}^{at}\int _{0}^{\pi }\left(\int _{0}^{2\pi }{\frac {d^{2}\varpi }{d\omega ^{2}}}d\omega \right){\frac {d\rho \,d\theta }{\rho \sin .\theta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51807f41b7777233e208dbe9db374f33d1870538)
On a évidemment
![{\displaystyle \int _{0}^{at}{\frac {d^{2}.\rho \varpi }{d\rho ^{2}}}d\rho ={\frac {d.at\varpi '}{ad\,t}}-f(x,y,z),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f083a92e98ed19f5889492ff590f7c8f198af2)
en désignant par
la valeur de
qui répond à
et observant que
est la valeur de
qui a lieu quand
On a aussi
![{\displaystyle \int _{0}^{\pi }{\frac {d.{\frac {d\varpi }{d\theta }}\sin .\theta }{d\theta }}d\theta =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f70ece84113a9a7abd7f04bd2fb33ed63ecc70)