on en conclura
![{\displaystyle \int _{0}^{\pi }\int _{0}^{2\pi }\Pi \left(x+at\cos .\theta ,y+at\sin .\theta \sin .\omega ,\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d92901cb2d2017b32da97d5f8fa17c3a80cc6f22)
![{\displaystyle \left.z+at\sin .\theta \cos .\omega \right)at\sin .\theta \,d\theta \,d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f70952d10e58e06c3cc95db129f6532da9e0d0)
![{\displaystyle =-\iiint \psi (x',y',z'){\frac {atdx'dy'dz'}{r'}},\ \ {\text{ou}}\ \ =-\iiint \psi (x',y',z')dx'dy'dz',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84cf5be06b88cfc3c97392d3afd9f6cf66538e37)
et, par conséquent
![{\displaystyle {\frac {d.}{dt}}\int _{0}^{\pi }\int _{0}^{2\pi }\Pi \left(x+at\cos .\theta ,y+at\sin .\theta \sin .\omega ,\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f348cb9a3fb9f4c84760d2ff3046ce362eec3d)
![{\displaystyle \left.z+at\sin .\theta \cos .\omega \right)t\sin .\theta \,d\theta \,d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99c4341f80b2773ab6f8de8cd7705a80081fe2f1)
![{\displaystyle =-\iiint \psi (x',y',z'){\frac {dx'dy'dz'}{r'}},\quad {\text{ou}}\quad =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e99caa62fd49b0a248acb1a1421a812094458ba)
selon qu’on aura
ou
Mais, d’après les équations (13), si l’on substitue
à
dans l’intégration relative à ces dernières variables, on aura
![{\displaystyle dx'dy'dz'=r'^{2}\sin .\theta 'dr'd\theta 'd\omega '\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54d0e232f9da83f965e9a741f628f98e8d8f9a26)
les limites relatives à
étant
celles qui répondent à
seront
et
on aura donc
![{\displaystyle \iiint \psi (x',y',z'){\frac {dx'dy'dz'}{r'}}=\int _{0}^{\infty }\int _{0}^{\pi }\int _{0}^{2\pi }\psi \left(x+r'\cos .\theta ',\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d84afd160347aa88e5ffc0866cc9352a546df9a5)
![{\displaystyle \left.y+r'\sin .\theta '\sin .\omega ',z+r'\sin .\theta '\cos .\omega '\right)r'\sin .\theta 'dr'd\theta 'd\omega ',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc41c74c6bdc42a99cbcf28782d7cdec4bd141f3)
et, par conséquent,
![{\displaystyle {\frac {d.}{dt}}\int _{0}^{\pi }\int _{0}^{2\pi }\Pi \left(x+at\cos .\theta ,y+at\sin .\theta \sin .\omega ,\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f348cb9a3fb9f4c84760d2ff3046ce362eec3d)
![{\displaystyle \left.z+at\sin .\theta \cos .\omega \right)t\sin .\theta \,d\theta \,d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99c4341f80b2773ab6f8de8cd7705a80081fe2f1)
![{\displaystyle =-\int _{at}^{\infty }\int _{0}^{\pi }\int _{0}^{2\pi }\psi \left(x+r'\cos .\theta ',y+r'\sin .\theta '\sin .\omega ',\right.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f297d363a731323bbcbb0290b3dbbb6cf2647ead)
![{\displaystyle \left.z+r'\sin .\theta '\cos .\omega '\right)r'\sin .\theta 'dr'd\theta 'd\omega '.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cb3c4f77480c95093a337858df9b15ffcd9195)