![{\displaystyle x=a+x_{1},\qquad y=b+y_{1},\qquad z=c+z_{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1424a755404ae65d7fbf9b00af545a86762c92c)
en désignant par
des fonctions de
nulles en même temps que
et convenablement déterminées, dès-lors nous aurons
![{\displaystyle {\begin{aligned}&\operatorname {d} 'x=\operatorname {d} 'x_{1}={\frac {\operatorname {d} x}{\operatorname {d} t}}+{\frac {\operatorname {d} x_{1}}{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} x_{1}}{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} x_{1}}{\operatorname {d} z}}\operatorname {d} 'z,\\\\&\operatorname {d} 'y=\operatorname {d} 'y_{1}={\frac {\operatorname {d} y}{\operatorname {d} t}}+{\frac {\operatorname {d} y_{1}}{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} y_{1}}{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} y_{1}}{\operatorname {d} z}}\operatorname {d} 'z,\\\\&\operatorname {d} 'z=\operatorname {d} 'z_{1}={\frac {\operatorname {d} z}{\operatorname {d} t}}+{\frac {\operatorname {d} z_{1}}{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} z_{1}}{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} z_{1}}{\operatorname {d} z}}\operatorname {d} 'z,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0d7b26654ad1623589776dd9ea8b6dd7a4d2be)
dans le cas actuel
étant très-petits, nous pouvons négliger les termes de plus d’une dimension par rapport à ces quantités ; nous aurons donc
![{\displaystyle \operatorname {d} 'x={\frac {\operatorname {d} x_{1}}{\operatorname {d} t}},\quad \operatorname {d} 'y={\frac {\operatorname {d} y_{1}}{\operatorname {d} t}},\quad \operatorname {d} 'z={\frac {\operatorname {d} z_{1}}{\operatorname {d} t}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b06cd72b17550c8c4136dc6e0e20926f4f621ab1)
et par suite, au moyen des équations (26)
![{\displaystyle {\frac {\operatorname {d} x_{1}}{\operatorname {d} t}}={\frac {\operatorname {d} \phi }{\operatorname {d} x}}+P,\quad {\frac {\operatorname {d} y_{1}}{\operatorname {d} t}}={\frac {\operatorname {d} \phi }{\operatorname {d} y}}+Q,\quad {\frac {\operatorname {d} z_{1}}{\operatorname {d} t}}={\frac {\operatorname {d} \phi }{\operatorname {d} z}}+R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa4dd09c28bed7490a3cb7f5fc9a220b82998a4)
d’où l’on tire, par l’élimination successive de
et ![{\displaystyle P,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd35af9d5901e795c83d9f519ac73264e74fa595)
![{\displaystyle {\begin{aligned}&{\frac {\operatorname {d} ^{2}x_{1}}{\operatorname {d} t\operatorname {d} y}}-{\frac {\operatorname {d} ^{2}y_{1}}{\operatorname {d} t\operatorname {d} x}}={\frac {\operatorname {d} P}{\operatorname {d} y}}-{\frac {\operatorname {d} Q}{\operatorname {d} x}},\\\\&{\frac {\operatorname {d} ^{2}y_{1}}{\operatorname {d} t\operatorname {d} z}}-{\frac {\operatorname {d} ^{2}z_{1}}{\operatorname {d} t\operatorname {d} y}}={\frac {\operatorname {d} Q}{\operatorname {d} z}}-{\frac {\operatorname {d} R}{\operatorname {d} y}},\\\\&{\frac {\operatorname {d} ^{2}z_{1}}{\operatorname {d} t\operatorname {d} x}}-{\frac {\operatorname {d} ^{2}x_{1}}{\operatorname {d} t\operatorname {d} z}}={\frac {\operatorname {d} R}{\operatorname {d} x}}-{\frac {\operatorname {d} P}{\operatorname {d} z}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f339bc8a7726d073cad294d08bb94e8d5cca2a16)
et de là en intégrant par rapport à ![{\displaystyle t,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6)