384
FRAGMENTS.
Il faudra donc ajouter ces termes aux trois équations (A), qui deviendront, par conséquent,
![{\displaystyle {\begin{alignedat}{5}&{\frac {d{\cfrac {\partial \mathrm {T} }{\partial p}}}{dt}}&+&q{\frac {\partial \mathrm {T} }{\partial r}}&-&r{\frac {\partial \mathrm {T} }{\partial q}}&+&\mathrm {\frac {3\left[(C-B)\mu \nu +F\left(\nu ^{2}-\mu ^{2}\right)+H\lambda \nu -G\mu \lambda \right]}{r^{5}}} &=&0,\\&{\frac {d{\cfrac {\partial \mathrm {T} }{\partial q}}}{dt}}&+&r{\frac {\partial \mathrm {T} }{\partial p}}&-&p{\frac {\partial \mathrm {T} }{\partial r}}&+&\mathrm {\frac {3\left[(A-C)\lambda \nu +G\left(\lambda ^{2}-\nu ^{2}\right)+F\lambda \mu -H\mu \nu \right]}{r^{5}}} &=&0,\\&{\frac {d{\cfrac {\partial \mathrm {T} }{\partial r}}}{dt}}&+&p{\frac {\partial \mathrm {T} }{\partial q}}&-&q{\frac {\partial \mathrm {T} }{\partial p}}&+&\mathrm {\frac {3\left[(B-A)\lambda \mu +H\left(\mu ^{2}-\nu ^{2}\right)+G\mu \nu -F\lambda \nu \right]}{r^{5}}} &=&0.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9bb991b017bac1607e1fcc697f54105601c47de)
De là nous tirerons les équations suivantes
![{\displaystyle {\begin{alignedat}{5}&{\frac {d\left(\xi '{\cfrac {\partial \mathrm {T} }{\partial p}}+\xi ''{\cfrac {\partial \mathrm {T} }{\partial q}}+\xi '''{\cfrac {\partial \mathrm {T} }{\partial r}}\right)}{dt}}&+&(\mathrm {P} )\xi '&+&(\mathrm {Q} )\xi ''&+&(\mathrm {R} )\xi '''&=&0,\\&{\frac {d\left(\eta '{\cfrac {\partial \mathrm {T} }{\partial p}}+\eta ''{\cfrac {\partial \mathrm {T} }{\partial q}}+\eta '''{\cfrac {\partial \mathrm {T} }{\partial r}}\right)}{dt}}&+&(\mathrm {P} )\eta '&+&(\mathrm {Q} )\eta ''&+&(\mathrm {R} )\eta '''&=&0,\\&{\frac {d\left(\zeta '{\cfrac {\partial \mathrm {T} }{\partial p}}+\zeta ''{\cfrac {\partial \mathrm {T} }{\partial q}}+\zeta '''{\cfrac {\partial \mathrm {T} }{\partial r}}\right)}{dt}}&+&(\mathrm {P} )\zeta '&+&(\mathrm {Q} )\zeta ''&+&(\mathrm {R} )\zeta '''&=&0,\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44acd58a9e7c10ac2fe0fbd779eba33b2f617015)
où
désignent les parties des précédentes qui ne dépendent pas de ![{\displaystyle p,\,q,\,r.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb82ac1574faa7c198d09988d8daf10daffd81d)
Faisons, pour abréger,
négligeons de plus, dans les premiers membres, les différences de
on aura alors
![{\displaystyle {\frac {\partial \mathrm {T} }{\partial p}}=\mathrm {A} p,\qquad {\frac {\partial \mathrm {T} }{\partial q}}=\mathrm {A} q,\qquad {\frac {\partial \mathrm {T} }{\partial r}}=\mathrm {A} r,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93eccab9b75e06d3f186cd977ec3968ff855c22a)
et nos équations deviendront
![{\displaystyle {\begin{alignedat}{3}&\mathrm {A} {\frac {d^{2}\mathrm {L} }{dt^{2}}}&+&\mathrm {\frac {3\left[(C-B)\mu \nu \xi '+(A-C)\lambda \nu \xi ''+(B-A)\lambda \mu \xi '''\right]}{r^{5}}} &=&0,\\&\mathrm {A} {\frac {d^{2}\mathrm {M} }{dt^{2}}}&+&\mathrm {\frac {3\left[(C-B)\mu \nu \eta '+(A-C)\lambda \nu \eta ''+(B-A)\lambda \mu \eta '''\right]}{r^{5}}} &=&0,\\&\mathrm {A} {\frac {d^{2}\mathrm {N} }{dt^{2}}}&+&\mathrm {\frac {3\left[(C-B)\mu \nu \zeta '+(A-C)\lambda \nu \zeta ''+(B-A)\lambda \mu \zeta '''\right]}{r^{5}}} &=&0.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/221ae469dbf65e924075d4c491e955ca014825e5)