et
comme les fonctions quelconques d’une variable primitive,
![{\displaystyle \mathrm {Z} '=\left({\frac {\mathrm {Z} '}{a'}}\right)a'+\left({\frac {\mathrm {Z} '}{b'}}\right)b'+\left({\frac {\mathrm {Z} '}{c'}}\right)c'+\ldots +\left({\frac {\mathrm {Z} '}{u'}}\right)u',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93eb2637b6fa7feb51071fa06975df90a9ef25be)
et, prenant de nouveau les fonctions dérivées,
![{\displaystyle {\begin{aligned}\mathrm {Z} ''=&\left({\frac {\mathrm {Z} '}{a'}}\right)a''+\left({\frac {\mathrm {Z} '}{b'}}\right)b''+\left({\frac {\mathrm {Z} '}{c'}}\right)c''+\ldots +\left({\frac {\mathrm {Z} '}{u'}}\right)u''\\&+\left({\frac {\mathrm {Z} ''}{a'^{2}}}\right)a'^{2}+2\left({\frac {\mathrm {Z} ''}{a'b'}}\right)a'b'+\left({\frac {\mathrm {Z} ''}{b'^{2}}}\right)b'^{2}+\ldots \\&+2\left({\frac {\mathrm {Z} ''}{a'u'}}\right)a'u'+2\left({\frac {\mathrm {Z} ''}{b'u'}}\right)b'u'+2\left({\frac {\mathrm {Z} ''}{c'u'}}\right)c'u'+\ldots +\left({\frac {\mathrm {Z} ''}{u'^{2}}}\right)u'^{2},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/678bd92c2271abf9f76d9149b8619bd6c6b6089e)
et ainsi de suite.
Donc, faisant
![{\displaystyle a'=0,\ \ a''=0,\ \ \ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f90c05a9329d65059c814bf358e773a3cd716c27)
![{\displaystyle b'={\frac {m-1}{m}}a,\ \ b''={\frac {m(m-1)}{m^{2}}},\ \ b'''=0,\ \ c'={\frac {m-2}{m}}b,\ \ \ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f10a04209b4b02fcba40e2915a456f8a0cccd364)
comme nous l’avons trouvé ci-dessus, on aura les équations
savoir
![{\displaystyle \left({\frac {\mathrm {Z} '}{a'}}\right)+\left({\frac {\mathrm {Z} '}{b'}}\right){\frac {m-1}{m}}a+\left({\frac {\mathrm {Z} '}{c'}}\right){\frac {m-2}{m}}b+\ldots +\left({\frac {\mathrm {Z} '}{u'}}\right){\frac {t}{m}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d15b7906655b11173a3844884b1c8eea7ffe21aa)
![{\displaystyle \left.{\begin{aligned}\left({\frac {\mathrm {Z} '}{b'}}\right)&{\frac {m(m-1)}{m^{2}}}+\left({\frac {\mathrm {Z} '}{c'}}\right){\frac {(m-1)(m-2)}{m^{2}}}a+\ldots +\left({\frac {\mathrm {Z} '}{u'}}\right){\frac {2s}{m^{2}}}\\+&\left({\frac {\mathrm {Z} ''}{a'^{2}}}\right)+2\left({\frac {\mathrm {Z} ''}{a'b'}}\right){\frac {m-1}{m}}a+\left({\frac {\mathrm {Z} ''}{b'^{2}}}\right)\left({\frac {m-1}{m}}a\right)^{2}+\ldots \\+&2\left[\left({\frac {\mathrm {Z} ''}{a'u'}}\right){\frac {1}{m}}+\left({\frac {\mathrm {Z} ''}{b'u'}}\right){\frac {m-1}{m^{2}}}a+\left({\frac {\mathrm {Z} ''}{c'uu'}}\right){\frac {m-2}{m^{2}}}b+\ldots \right]t\\+&\left({\frac {\mathrm {Z} ''}{u'^{2}}}\right){\frac {t^{2}}{m^{2}}}+\ldots \end{aligned}}\right\}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4545da44942f0783e89407eb4f172ca8a613d46e)
et ainsi de suite, dans lesquelles les fonctions dérivées partielles
![{\displaystyle \left({\frac {\mathrm {Z} '}{a'}}\right),\quad \left({\frac {\mathrm {Z} '}{b'}}\right),\quad \ldots ,\quad \left({\frac {\mathrm {Z} ''}{a'^{2}}}\right),\quad \left({\frac {\mathrm {Z} ''}{a'b'}}\right),\quad \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/245bb15fbfa3dfa008798b99735171736a128a3f)
seront des fonctions connues de ![{\displaystyle a,b,c,\ldots ,u.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a071accf76600ab2a229716bea277b5549dfab56)