En différentiant de nouveau, il vient
![{\displaystyle {\begin{aligned}&{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}={\frac {4k^{2}p^{2}}{\left\{(a-Vt\operatorname {Cos} .\alpha )^{2}-4k^{2}\left({\frac {p}{a}}\right)^{2}t^{2}\right\}^{\frac {3}{2}}}}=-{\frac {4k^{2}p^{2}}{(x-a)^{3}}},\\\\&{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}={\frac {4k^{2}q^{2}}{\left\{(b-Vt\operatorname {Cos} .\beta )^{2}-4k^{2}\left({\frac {q}{b}}\right)^{2}t^{2}\right\}^{\frac {3}{2}}}}=-{\frac {4k^{2}q^{2}}{(y-b)^{3}}},\\\\&{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}={\frac {4k^{2}r^{2}}{\left\{(c-Vt\operatorname {Cos} .\gamma )^{2}-4k^{2}\left({\frac {r}{c}}\right)^{2}t^{2}\right\}^{\frac {3}{2}}}}=-{\frac {4k^{2}r^{2}}{(z-c)^{3}}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdaafd964a088456c9b4c1354bb54521637a696)
on aura donc aussi (16)
![{\displaystyle 2k^{2}P=-{\frac {4k^{2}p^{2}}{(x-a)^{3}}},\quad 2k^{2}Q=-{\frac {4k^{2}q^{2}}{(y-b)^{3}}},\quad 2k^{2}R=-{\frac {4k^{2}r^{2}}{(z-c)^{3}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8656bfed73ee84533d91b10043b7a5b7e045bd)
c’est-à-dire,
![{\displaystyle \left({\frac {\operatorname {d} u}{\operatorname {d} x}}\right)=P=-{\frac {2p^{2}}{(x-a)^{3}}},\quad \left({\frac {\operatorname {d} u}{\operatorname {d} y}}\right)=Q=-{\frac {2q^{2}}{(y-b)^{3}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3884c615361bd6c2bb0d53be6a8ac72ce8bf08)
![{\displaystyle \left({\frac {\operatorname {d} u}{\operatorname {d} z}}\right)=R=-{\frac {2r^{2}}{(z-c)^{3}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dedd472fd3459ff5b1e14921cd80d71ad4280b95)
d’où,
![{\displaystyle \operatorname {d} u=-{\frac {2p^{2}}{(x-a)^{3}}}-{\frac {2q^{2}}{(y-b)^{3}}}-{\frac {2r^{2}}{(z-c)^{3}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f02d7e55701a358ba4d94df5bf7c1d51ef34253)
et, par suite, en intégrant,
![{\displaystyle u=\left({\frac {p}{x-a}}\right)^{2}+\left({\frac {q}{y-b}}\right)^{2}+\left({\frac {r}{z-c}}\right)^{2}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69bad5b95290eb140a644312e311655572bd0139)
telle est donc la définition du milieu dont il s’agit. Nous n’ajoutons point de constante, attendu qu’en augmentant ou en diminuant, d’une même quantité, la densité de tous les points du milieu, on ne change rien aux circonstances du phénomène.