et
si
, et
deviennent respectivement
et
et
deviendront respectivement (48)
![{\displaystyle {\begin{alignedat}{2}P+{\frac {\operatorname {d} P}{\operatorname {d} x}}{\frac {g}{1}}&+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+\ldots &Q+{\frac {\operatorname {d} Q}{\operatorname {d} x}}{\frac {g}{1}}&+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+\ldots &\\\\{\frac {\operatorname {d} P}{\operatorname {d} y}}{\frac {h}{1}}&+2{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+\ldots &{\frac {\operatorname {d} Q}{\operatorname {d} y}}{\frac {h}{1}}&+2{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+\ldots &\\\\&+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots &&+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots \,;&\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee81668b26a09d21ed78a410247a7a709b27f4a)
de sorte qu’en dénotant par
et
les accroissement respectifs de
et
on aura
![{\displaystyle G={\frac {\operatorname {d} P}{\operatorname {d} x}}{\frac {g}{1}}+{\frac {\operatorname {d} P}{\operatorname {d} y}}{\frac {h}{1}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a725adb04e1a862d1af92230e25a45637b860ba6)
![{\displaystyle H={\frac {\operatorname {d} Q}{\operatorname {d} x}}{\frac {g}{1}}+{\frac {\operatorname {d} Q}{\operatorname {d} y}}{\frac {h}{1}}+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea39328eebad81165d4d05f501909b93da073aea)
mais,
et
devenant respectivement
et
doit devenir (48)
![{\displaystyle {\begin{alignedat}{1}S+{\frac {\operatorname {d} S}{\operatorname {d} P}}{\frac {G}{1}}&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} P^{2}}}{\frac {G^{2}}{1.2}}+\ldots \\\\+{\frac {\operatorname {d} S}{\operatorname {d} Q}}{\frac {H}{1}}&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} P\operatorname {d} Q}}{\frac {GH}{1.2}}+\ldots \\\\&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} Q^{2}}}{\frac {H^{2}}{1.2}}+\ldots \end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e14ab59f2c176601d1d518f2ccc39fd66f9d859)
ou, en mettant pour
et
leurs valeurs, développant et ordonnant,
![{\displaystyle S+\left({\frac {\operatorname {d} S}{\operatorname {d} P}}{\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {\operatorname {d} S}{\operatorname {d} Q}}{\frac {\operatorname {d} Q}{\operatorname {d} x}}\right){\frac {g}{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb6492e26f8d55a83cf49d7ed670dbb175df3635)