![{\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} x}}\operatorname {d} x=P\operatorname {d} p+Q\operatorname {d} q+R\operatorname {d} r+\ldots +p{\frac {\operatorname {d} P}{\operatorname {d} x}}\operatorname {d} x-p{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}\operatorname {d} x+p{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{3}}}\operatorname {d} x-\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbc329a358ad75011efff434c6f6505e191795b)
et par conséquent
![{\displaystyle V=\int P\operatorname {d} p+\int Q\operatorname {d} q+\int R\operatorname {d} r+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd7619bda080e693c58597f10c3d848496cb5a6)
![{\displaystyle +\int p{\frac {\operatorname {d} P}{\operatorname {d} x}}\operatorname {d} x-\int p{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}\operatorname {d} x+\int p{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{3}}}\operatorname {d} x-\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/26fed03d9cbcbf6d5eb78d4b67a7f00e905d57a2)
Or, en intégrant par parties, on trouve
![{\displaystyle {\begin{aligned}&\int P\operatorname {d} p=Pp-\int p{\frac {\operatorname {d} P}{\operatorname {d} x}}\operatorname {d} x,\\\\&\int Q\operatorname {d} q=Qq-p{\frac {\operatorname {d} Q}{\operatorname {d} x}}+\int p{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}\operatorname {d} x,\\\\&\int R\operatorname {d} r=Rr-q{\frac {\operatorname {d} R}{\operatorname {d} x}}+p{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}-\int p{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{3}}}\operatorname {d} x,\\&\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/43dbc32c79f972b7af2a6a9c129827f2a34598da)
ce qui donne, en subtituant,
![{\displaystyle V=p\left(P-{\frac {\operatorname {d} Q}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}-\ldots \right)+q\left(Q-{\frac {\operatorname {d} R}{\operatorname {d} x}}+\ldots \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0efaccf56cf29defc03c1562f1840e6fc2056f)
![{\displaystyle +r(R-\ldots )+\ldots +C\,;\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/390b24f5b5104458cf006489447a5c9a333367ca)
(D)
qui n’est plus que de l’ordre ![{\displaystyle 2n-1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62f05639a38051132031ad62fb4923d9b63e1ccc)
3.o Quand
ne contient ni
ni
on peut abaisser l’équation (A), qu’on appelle quelquefois l’équation indéfinie, parce que son intégrale donne l’équation du maximum ou du minimum, avec des constantes arbitraires. On a alors, à la fois,
![{\displaystyle M=0,\qquad N=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c584f9f955b219343e49ce7761ddb27caeebc48)