Retranchant la différentielle complète
![{\displaystyle {\frac {p\,dp+q\,dq+r\,dr}{dx}}dx+{\frac {p\,dp+q\,dq+r\,dr}{dy}}dy+{\frac {p\,dp+q\,dq+r\,dr}{dz}}dz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/957760da5ce98a7c60c20899e8071ac192adba9f)
on aura la quantité
![{\displaystyle {\begin{aligned}{\frac {dp}{dt}}&dx+{\frac {dq}{dt}}dy+{\frac {dr}{dt}}dz+\left({\frac {dp}{dy}}-{\frac {dq}{dx}}\right)(qdx-pdy)\\&+\left({\frac {dp}{dz}}-{\frac {dr}{dx}}\right)(rdx-pdz)+\left({\frac {dq}{dz}}-{\frac {dr}{dy}}\right)(rdy-qdz),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773528ffb7b1b43bd097e9a8d581d912307f78dc)
qui devra être une différentielle complète.
Soient
les valeurs de
dans un instant quelconque où
il est clair que pour
(
étant fort petit) on aura
![{\displaystyle {\begin{aligned}p=&p'+p''\theta +p'''\theta ^{2}+\ldots ,\\q=&q'+q''\theta \,+q'''\theta ^{2}+\ldots ,\\r=&r'\,+r''\theta +r'''\theta ^{2}+\ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8db2e99a674dc88bf5514e0824e5cc7182832ec6)
étant des fonctions de
et de la quantité
qu’on regarde maintenant comme constante. Faisant ces substitutions dans la quantité précédente et prenant
on aura une transformée de cette forme,
![{\displaystyle \alpha =\beta \theta +\gamma \theta ^{2}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ef4cc69ebb3046372264918d3acd9e48d1e7a0)
en supposant
![{\displaystyle {\begin{aligned}\alpha =&p''dx+q''dy+r''dz\\&+\left({\frac {dp'}{dy}}-{\frac {dq'}{dx}}\right)(q'dx-p'dy)+\left({\frac {dp'}{dz}}-{\frac {dr'}{dx}}\right)(r'dx-p'dz)\\&+\left({\frac {dq'}{dz}}-{\frac {dr'}{dy}}\right)(r'dy-q'dz),\\\\\beta =&2(p'''dx+q'''dy+r'''dz)\\&+\left({\frac {dp'}{dy}}-{\frac {dq'}{dx}}\right)(q''dx-p''dy)+\left({\frac {dp''}{dy}}-{\frac {dq''}{dx}}\right)(q'dx-p'dy)\\&+\left({\frac {dp'}{dz}}-{\frac {dr'}{dx}}\right)(r''dx-p''dz)\,+\left({\frac {dp''}{dz}}-{\frac {dr''}{dx}}\right)(r'dx-p'dz)\\&+\left({\frac {dq'}{dz}}-{\frac {dr'}{dy}}\right)(r''dy-q''dz)\;+\left({\frac {dq''}{dz}}-{\frac {dr''}{dy}}\right)(r'dy-q'dz),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2298fca93870cdda0e6039b45c5bf9dfc1d162dc)