16. Supposons maintenant que
soit une fonction de deux variables
et
on aura dans ce cas, en faisant
![{\displaystyle \Delta 'u=\left(1+\Delta _{\xi }u\right)^{\frac {\xi '}{\xi }}\left(1+\Delta _{\psi }u\right)^{\frac {\psi '}{\psi }}-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d97c04572a8162f251e6b1dafb238bdd3afba2)
La quantité
donne comme ci-dessus la série
![{\displaystyle 1+{\frac {\xi '}{\xi }}\Delta _{\xi }u+{\frac {\xi '(\xi '-\xi )}{2\xi ^{2}}}\Delta _{\xi ^{2}}^{2}u+{\frac {\xi '(\xi '-\xi )(\xi '-2\xi )}{2.3.\xi ^{3}}}\Delta _{\xi ^{3}}^{3}u+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea5f12f7c0c234f0ee911d648f0b1d3995e1bde)
et de même la quantité
donnera la série
![{\displaystyle 1+{\frac {\psi '}{\psi }}\Delta _{\psi }u+{\frac {\psi '(\psi '-\psi )}{2\psi ^{2}}}\Delta _{\psi ^{2}}^{2}u+{\frac {\psi '(\psi '-\psi )(\psi '-2\psi )}{2.3.\psi ^{3}}}\Delta _{\psi ^{3}}^{3}u+\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11519b0a3fde628db13d202d2413d254b648fb6e)
Donc, multipliant une série par l’autre et ayant égard aux remarques faites vers la fin du no 13, on aura
![{\displaystyle {\begin{aligned}1&+{\frac {\xi '}{\xi }}\Delta _{\xi }u+{\frac {\psi '}{\psi }}\Delta _{\psi }u\\&+{\frac {\xi '(\xi '-\xi )}{2\xi ^{2}}}\Delta _{\xi ^{2}}^{2}u+{\frac {\xi '\psi '}{\xi \psi }}\Delta _{\xi \psi }^{2}u+{\frac {\psi '(\psi '-\psi )}{2\psi ^{2}}}\Delta _{\psi ^{2}}^{2}u\\&+{\frac {\xi '(\xi '-\xi )(\xi '-2\xi )}{2.3.\xi ^{3}}}\Delta _{\xi ^{3}}^{3}u+{\frac {\xi '(\xi '-\xi )}{2\xi ^{2}}}{\frac {\psi '}{\psi }}\Delta _{\xi ^{2}\psi }^{3}u\\&+{\frac {\psi '(\psi '-\psi )}{2\psi ^{2}}}{\frac {\xi '}{\xi }}\Delta _{\psi ^{2}\xi }^{3}u+{\frac {\psi '(\psi '-\psi )(\psi '-2\psi )}{2.3.\psi ^{3}}}\Delta _{\psi ^{3}}^{3}u\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad89fce44353a9ebab330a24458d7a9924adce8)
Donc
![{\displaystyle {\begin{aligned}\Delta 'u&={\frac {\xi '}{\xi }}\Delta _{\xi }u+{\frac {\psi '}{\psi }}\Delta _{\psi }u\\&+{\frac {\xi '(\xi '-\xi )}{2\xi ^{2}}}\Delta _{\xi ^{2}}^{2}u+{\frac {\xi '\psi '}{\xi \psi }}\Delta _{\xi \psi }^{2}u+{\frac {\psi '(\psi '-\psi )}{2\psi ^{2}}}\Delta _{\psi ^{2}}^{2}u\\&+{\frac {\xi '(\xi '-\xi )(\xi '-2\xi )}{2.3.\xi ^{3}}}\Delta _{\xi ^{3}}^{3}u+{\frac {\xi '(\xi '-\xi )}{2\xi ^{2}}}{\frac {\psi '}{\psi }}\Delta _{\xi ^{2}\psi }^{3}u\\&+{\frac {\psi '(\psi '-\psi )}{2\psi ^{2}}}{\frac {\xi '}{\xi }}\Delta _{\psi ^{2}\xi }^{3}u+{\frac {\psi '(\psi '-\psi )(\psi '-2\psi )}{2.3.\psi ^{3}}}\Delta _{\psi ^{3}}^{3}u\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2421c5ef09d952b762ee6f12a1918f5a9092d27)
c’est l’accroissement que doit prendre la fonction
lorsque
et
y deviennent à la fois ![{\displaystyle x+\xi ',\ y+\psi '.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53753925940c415f4f5c253c8516035269cfdef5)