en faisant, pour abréger,
![{\displaystyle f(x+\rho \cos .\theta ,y+\rho \sin .\theta \sin .\omega ,z+\rho \sin .\theta \cos .\omega )=\varpi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc474fdab1141520a7d2aa7ab81cc32509c61cd)
D’après cette expression de
on aura
![{\displaystyle {\begin{aligned}{\frac {d\varpi }{d\rho }}&={\frac {d\varpi }{dx}}\cos .\theta +{\frac {d\varpi }{dy}}\sin .\theta \sin .\omega +{\frac {d\varpi }{dz}}\sin .\theta \cos .\omega ,\\{\frac {d\varpi }{d\rho }}&=-{\frac {d\varpi }{dx}}\rho \sin .\theta +{\frac {d\varpi }{dy}}\rho \cos .\theta \sin .\omega +{\frac {d\varpi }{dz}}\rho \cos .\theta \cos .\omega ,\\{\frac {d\varpi }{d\omega }}&={\frac {d\varpi }{dy}}\rho \sin .\theta \cos .\omega -{\frac {d\varpi }{dz}}\rho \sin .\theta \sin .\omega \,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51b408dbec50b76da413f9221737141412e98aa2)
d’où l’on tire
![{\displaystyle {\begin{aligned}{\frac {d\varpi }{dx}}&={\frac {d\varpi }{d\rho }}\cos .\theta -{\frac {d\varpi }{d\theta }}{\frac {\sin .\theta }{\rho }},\\{\frac {d\varpi }{dy}}&={\frac {d\varpi }{d\rho }}\sin .\theta \sin .\omega +{\frac {d\varpi }{d\theta }}{\frac {\cos .\theta \sin .\omega }{\rho }}+{\frac {d\varpi }{d\omega }}{\frac {\cos .\omega }{\rho \sin .\theta }},\\{\frac {d\varpi }{dz}}&={\frac {d\varpi }{d\rho }}\sin .\theta \cos .\omega +{\frac {d\varpi }{d\theta }}{\frac {\cos .\theta \cos .\omega }{\rho }}-{\frac {d\varpi }{d\omega }}{\frac {\sin .\omega }{\rho \sin .\theta }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fedbbc1e19d2d5a47f9133c0dd796d366b5d2e41)
En mettant
au lieu de
dans la première de ces trois dernières équations, on a
![{\displaystyle {\frac {d^{2}\varpi }{dx^{2}}}={\frac {d^{2}\varpi }{dx\,d\rho }}\cos .\theta -{\frac {d^{2}\varpi }{dx\,d\theta }}{\frac {\sin .\theta }{\rho }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f89e33bc1446108afec12d0b50961230833dc5)
et en différentiant successivement cette même équation par rapport à
et à
il vient
![{\displaystyle {\begin{aligned}{\frac {d^{2}\varpi }{dx\,d\rho }}&={\frac {d^{2}\varpi }{d\rho ^{2}}}\cos .\theta -{\frac {d^{2}\varpi }{d\rho \,d\theta }}{\frac {\sin .\theta }{\rho }}+{\frac {d\varpi }{d\theta }}{\frac {\sin .\theta }{\rho ^{2}}},\\{\frac {d^{2}\varpi }{dx\,d\theta }}&={\frac {d^{2}\varpi }{d\rho \,d\theta }}\cos .\theta -{\frac {d\varpi }{d\rho }}\sin .\theta -{\frac {d^{2}\varpi }{d\theta ^{2}}}{\frac {\sin .\theta }{\rho }}-{\frac {d\varpi }{d\theta }}{\frac {\cos .\theta }{\rho }}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e6fce51b94f4970c90e9f8a1bcb3cefb0565b3)
par conséquent, on aura