252
QUESTIONS
mais, parce que
et
sont, l’un et l’autre, des fonctions de
et
on doit avoir aussi
![{\displaystyle \mathrm {d} x={\tfrac {\mathrm {d} x}{\mathrm {d} u}}\mathrm {d} u+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}\mathrm {d} v,\quad \mathrm {d} y={\tfrac {\mathrm {d} y}{\mathrm {d} u}}\mathrm {d} u+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}\mathrm {d} v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0876f46c350185777d4071cc4950865402f4a905)
;
substituant donc dans l’équation précédente, elle deviendra
![{\displaystyle \mathrm {(\mathrm {I} )} \quad \left\{{\tfrac {\mathrm {d} x}{\mathrm {d} u}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}q-{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\right\}\mathrm {d} u+\left\{{\tfrac {\mathrm {d} x}{\mathrm {d} v}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}q-{\tfrac {\mathrm {d} z}{\mathrm {d} v}}\right\}\mathrm {d} v=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba0cc47a33eebdfe8fc02b43d3528790c3c91c0)
.
La différentielle complète de cette équation, par rapport à
et
sera
![{\displaystyle (\mathrm {II} )\left\{{\begin{array}{ll}\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} u}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right)^{2}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u^{2}}}\right\}\mathrm {d} u^{2}\\+2\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u\mathrm {d} v}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u\mathrm {d} v}}q+{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}r+\left[{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right]s+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u\mathrm {d} v}}\right\}\mathrm {d} u\mathrm {d} v\\+\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} v^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} v^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} v}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} v}}\right)^{2}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} v^{2}}}\right\}\mathrm {d} v^{2}=0;\\\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/430fac8701736d114fa5ad59215540cefa9d7f30)
or, à cause de l’indépendance des différentielles
, les équations
et
se partagent dans les cinq suivantes :
![{\displaystyle (1)\quad {\tfrac {\mathrm {d} x}{\mathrm {d} u}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}q={\tfrac {\mathrm {d} z}{\mathrm {d} u}},\quad (2)\quad {\tfrac {\mathrm {d} x}{\mathrm {d} v}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}q={\tfrac {\mathrm {d} z}{\mathrm {d} v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b059d168d6bb44950f49bddbf561f54bd458a6bb)
,
![{\displaystyle (3)\quad {\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} u}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right)^{2}t={\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed9329290751fc23a82a4d6f7eebb0bd9d60878)
,
![{\displaystyle (4)\quad {\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u\mathrm {d} v}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u\mathrm {d} v}}q+{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}r+\left[{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right]s+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}t={\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u\mathrm {d} v}}{\text{,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/454ffd29e60d314d424fbb610dfa9ce74e5da0fe)
[1].
- ↑ Ces équations, en y changeant x et y en u et v, et vice versa, rentrent dans celles qu’a données M. Lacroix, pour une transformation analogue à celle-ci ; mais