Page:Mémoires de l’Académie des sciences, Tome 22.djvu/792

(2)

$\rho _{_{'}}=\mathrm {\frac {x_{_{'}}(\xi _{_{'}}-\xi \Delta _{_{'}}\xi _{_{'}})+y_{_{'}}(\eta _{_{'}}-\eta \Delta _{_{'}}\eta _{_{'}})+z_{_{'}}(\zeta _{_{'}}-\zeta \Delta _{_{'}}\zeta _{_{'}})}{r_{_{'}}}} \,;$

et les formules (2), (5) du même paragraphe donneront

(3)

$\left\{{\begin{alignedat}{4}\operatorname {D} _{t}^{2}\xi &=&\mathrm {L} \xi &+&\mathrm {R} \eta &+&\mathrm {Q} \zeta &+&\mathrm {L} _{_{'}}\xi _{_{'}}&+&\mathrm {R} _{_{'}}\eta _{_{'}}&+&\mathrm {Q} _{_{'}}\zeta _{_{'}},\\\operatorname {D} _{t}^{2}\eta &=&\mathrm {R} \xi &+&\mathrm {M} \eta &+&\mathrm {P} \zeta &+&\mathrm {R} _{_{'}}\xi _{_{'}}&+&\mathrm {M} _{_{'}}\eta _{_{'}}&+&\mathrm {P} _{_{'}}\zeta _{_{'}},\\\operatorname {D} _{t}^{2}\zeta &=&\mathrm {Q} \xi &+&\mathrm {P} \eta &+&\mathrm {N} \zeta &+&\mathrm {Q} _{_{'}}\xi _{_{'}}&+&\mathrm {P} _{_{'}}\eta _{_{'}}&+&\mathrm {N} _{_{'}}\zeta _{_{'}},\end{alignedat}}\right.$

les valeurs de

${\begin{array}{llllll}\mathrm {L} ,&\mathrm {M} ,&\mathrm {N} ,&\mathrm {P} ,&\mathrm {Q} ,&\mathrm {R} ,\\\mathrm {L} _{_{'}},&\mathrm {M} _{_{'}},&\mathrm {N} _{_{'}},&\mathrm {P} _{_{'}},&\mathrm {Q} _{_{'}},&\mathrm {R} _{_{'}},\end{array}}$

étant déterminées par les formules symboliques

${\begin{aligned}\mathrm {L} \,&=\mathbf {S} m\left[f(\mathrm {r} )+\mathrm {\frac {x^{2}}{r}} \operatorname {D} _{\mathrm {r} }f(\mathrm {r} )\right]\Delta -\mathbf {S} m_{_{'}}\left[\operatorname {f} (\mathrm {r} _{_{'}})+\mathrm {\frac {x_{_{'}}^{2}}{r_{_{'}}}} \operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}})\right],{\text{ etc.}}\\\mathrm {P} \,&=\mathbf {S} m\mathrm {\frac {yz}{r}} \operatorname {D} _{\mathrm {r} }f(\mathrm {r} )\Delta -\mathbf {S} m_{_{'}}\mathrm {\frac {y_{_{'}}z_{_{'}}}{r_{_{'}}}} \operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}})\Delta ,{\text{ etc.}}\\\mathrm {L} _{_{'}}&=\mathbf {S} m_{_{'}}\left[\operatorname {f} (\mathrm {r} _{_{'}})+\mathrm {\frac {x_{_{'}}^{2}}{r_{_{'}}}} \operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}})\right](1+\Delta _{_{'}}),{\text{ etc.}}\\\mathrm {P} _{_{'}}&=\mathbf {S} m_{_{'}}\mathrm {\frac {y_{_{'}}z_{_{'}}}{r_{_{'}}}} \operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}})(1+\Delta _{_{'}}),{\text{ etc.}}\end{aligned}}$

maintenant on pose, comme dans le paragraphe précédent,

$u=\operatorname {D} _{x},\qquad v=\operatorname {D} _{y},\qquad w=\operatorname {D} _{z},$

$\iota =x\operatorname {D} _{x}+y\operatorname {D} _{y}+z\operatorname {D} _{z},\qquad \iota _{_{'}}=x_{_{'}}\operatorname {D} _{x}+y_{_{'}}\operatorname {D} _{y}+z_{_{'}}\operatorname {D} _{z},$

alors, en ayant égard aux formules symboliques

$\Delta =e^{\iota }-1,\qquad 1+\Delta _{_{'}}=e^{\iota _{_{'}}},$

et en posant d’ailleurs

(4)

$\left\{{\begin{aligned}\mathrm {G} &=\mathbf {S} mf(\mathrm {r} )\left(e^{\iota }-1\right)-\mathbf {S} m_{_{'}}\operatorname {f} (\mathrm {r} _{_{'}}),\\\mathrm {H} &=\mathbf {S} {\frac {m}{\mathrm {r} }}\operatorname {D} _{\mathrm {r} }f(\mathrm {r} )\left(e^{\iota }-{\frac {\iota ^{2}}{2}}\right)-\mathbf {S} {\frac {m_{_{'}}}{\mathrm {r} _{_{'}}}}\operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}}){\frac {\iota _{_{'}}^{2}}{2}},\\\end{aligned}}\right.$

(5)

$\left\{{\begin{aligned}\mathrm {G} _{_{'}}&=\mathbf {S} m_{_{'}}\operatorname {f} (\mathrm {r} _{_{'}})e^{\iota _{_{'}}},\\\mathrm {H} _{_{'}}&=\mathbf {S} {\frac {m_{_{'}}}{\mathrm {r} _{_{'}}}}\operatorname {D} _{\mathrm {r} _{_{'}}}\operatorname {f} (\mathrm {r} _{_{'}})e^{\iota _{_{'}}},\qquad \qquad \qquad \qquad \qquad \ \ \end{aligned}}\right.$