Ensuite, en regardant
comme très-petite et faisant dans la différentielle de
![{\displaystyle {\begin{alignedat}{3}dx=&z\beta ,&dx'=&z'\beta ,&dx''=&z''\beta ,\ldots ,\\dz=&-x\beta ,\qquad &dz'=&-x'\beta ,\qquad &dz''=&-x''\beta ,\ldots ,\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28e0bc711628a2f156023236d9f2b31827537868)
on aura par un raisonnement semblable l’équation
![{\displaystyle 0={\frac {d\Omega }{dx}}z+{\frac {d\Omega }{dx'}}z'+{\frac {d\Omega }{dx''}}z''+\ldots -{\frac {d\Omega }{dz}}x-{\frac {d\Omega }{dz'}}x'-{\frac {d\Omega }{dz''}}x''-\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc877cfa5cd47907e137f9546ed69dc18294ee22)
Enfin l’on fera, dans l’hypothèse de
très-petite,
![{\displaystyle {\begin{alignedat}{3}dy=&z\gamma ,&dy'=&z'\gamma ,&dy''=&z''\gamma ,\ldots ,\\dz=&-y\gamma ,\qquad &dz'=&-y'\gamma ,\qquad &dz''=&-y''\gamma ,\ldots ,\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d3f62b35090fa979274c36b8d3997f879c9a8c1)
et l’on aura
![{\displaystyle 0={\frac {d\Omega }{dy}}z+{\frac {d\Omega }{dy'}}z'+{\frac {d\Omega }{dy''}}z''+\ldots -{\frac {d\Omega }{dz}}y-{\frac {d\Omega }{dz'}}y'-{\frac {d\Omega }{dz''}}y''-\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab223a4c0f6dad0c44c015526a8aa171d4472c98)
7. Si maintenant on substitue dans ces trois équations les valeurs de
que donnent les équations différentielles du no 2, on aura ces trois-ci
![{\displaystyle {\begin{aligned}0&=\mathrm {M} y{\frac {d^{2}x}{dt^{2}}}+\mathrm {M} 'y'{\frac {d^{2}x'}{dt^{2}}}+\mathrm {M} ''y''{\frac {d^{2}x''}{dt^{2}}}+\ldots \\&-\mathrm {M} x{\frac {d^{2}y}{dt^{2}}}-\mathrm {M} 'x'{\frac {d^{2}y'}{dt^{2}}}-\mathrm {M} ''x''{\frac {d^{2}y''}{dt^{2}}}-\ldots ,\\0&=\mathrm {M} z{\frac {d^{2}x}{dt^{2}}}+\mathrm {M} 'z'{\frac {d^{2}x'}{dt^{2}}}+\mathrm {M} ''z''{\frac {d^{2}x''}{dt^{2}}}+\ldots \\&-\mathrm {M} x{\frac {d^{2}z}{dt^{2}}}-\mathrm {M} 'x'{\frac {d^{2}z'}{dt^{2}}}-\mathrm {M} ''x''{\frac {d^{2}z''}{dt^{2}}}-\ldots ,\\0&=\mathrm {M} z{\frac {d^{2}y}{dt^{2}}}+\mathrm {M} 'z'{\frac {d^{2}y'}{dt^{2}}}+\mathrm {M} ''z''{\frac {d^{2}y''}{dt^{2}}}+\ldots \\&-\mathrm {M} y{\frac {d^{2}z}{dt^{2}}}-\mathrm {M} 'y'{\frac {d^{2}z'}{dt^{2}}}-\mathrm {M} ''y''{\frac {d^{2}z''}{dt^{2}}}-\ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25aea5182e5bfcf741d64c59446dfd464dd1d957)