On a de plus, par le no 64,
![{\displaystyle xdy-ydx=cdt,\qquad xdz-zdx=c'dt,\qquad ydz-zdy=c''dt\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7183c898cc1fca84cff11b9694cf88cc05274a2)
les équations différentielles en
deviendront ainsi
![{\displaystyle {\begin{aligned}\operatorname {f} =&-dy{\frac {\partial R}{\partial v}}-dz\left[\left(1+s^{2}\right)\cos v{\frac {\partial {\rm {R}}}{\partial s}}-rs\cos v{\frac {\partial {\rm {R}}}{\partial r}}+s\sin v{\frac {\partial {\rm {R}}}{\partial v}}\right]\\\\&-cdt\left(\sin v{\frac {\partial {\rm {R}}}{\partial r}}+{\frac {\cos v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}-{\frac {s\sin v}{r}}{\frac {\partial {\rm {R}}}{\partial s}}\right)-{\frac {c'dt}{r}}{\frac {\partial {\rm {R}}}{\partial s}},\\\\\operatorname {f} '=&dx{\frac {\partial R}{\partial v}}-dz\left[\left(1+s^{2}\right)\sin v{\frac {\partial {\rm {R}}}{\partial s}}-rs\sin v{\frac {\partial {\rm {R}}}{\partial r}}-s\cos v{\frac {\partial {\rm {R}}}{\partial v}}\right]\\\\&+cdt\left(\cos v{\frac {\partial {\rm {R}}}{\partial r}}-{\frac {\sin v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}-{\frac {s\cos v}{r}}{\frac {\partial {\rm {R}}}{\partial s}}\right)-{\frac {c''dt}{r}}{\frac {\partial {\rm {R}}}{\partial s}},\\\\\operatorname {f} ''=&dx\left[\left(1+s^{2}\right)\cos v{\frac {\partial {\rm {R}}}{\partial s}}-rs\cos v{\frac {\partial {\rm {R}}}{\partial r}}+s\sin v{\frac {\partial {\rm {R}}}{\partial v}}\right]\\\\&dy\left[\left(1+s^{2}\right)\sin v{\frac {\partial {\rm {R}}}{\partial s}}-rs\sin v{\frac {\partial {\rm {R}}}{\partial r}}-s\cos v{\frac {\partial {\rm {R}}}{\partial v}}\right]\\\\&+c'dt\left(\cos v{\frac {\partial {\rm {R}}}{\partial r}}-{\frac {\sin v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}-{\frac {s\cos v}{r}}{\frac {\partial {\rm {R}}}{\partial s}}\right)\\\\&+c''dt\left(\sin v{\frac {\partial {\rm {R}}}{\partial r}}+{\frac {\cos v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}-{\frac {s\sin v}{r}}{\frac {\partial {\rm {R}}}{\partial s}}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2424cbc485293e82e2047ef2bcaf9378eee4bd97)
Les quantités
dépendent, comme on l’a vu dans le no 64, de l’inclinaison de l’orbite de
sur le plan fixe, en sorte que ces quantités se réduiraient à zéro, si cette inclinaison était nulle ; d’ailleurs, il est aisé de voir, par la nature de
que
est de l’ordre des inclinaisons des orbites ; en négligeant donc les carrés et les produits de ces inclinaisons, les expressions précédentes de
et de
deviendron
![{\displaystyle {\begin{aligned}\operatorname {f} &=-dy{\frac {\partial R}{\partial v}}-cdt\left(\sin v{\frac {\partial {\rm {R}}}{\partial r}}+{\frac {\cos v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}\right),\\\\\operatorname {f} '&=\ \ dx{\frac {\partial R}{\partial v}}+cdt\left(\cos v{\frac {\partial {\rm {R}}}{\partial r}}-{\frac {\sin v}{r}}{\frac {\partial {\rm {R}}}{\partial v}}\right)\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5cedd1efb8fa745f231ed1e96d99eafec3154c0)
or on a
![{\displaystyle dx=d(r\cos v),\quad dy=d(r\sin v),\quad cdt=xdy-ydx=r^{2}dv\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f804a514f7ec1f6ec038ec64c4b20bbf8b42c3)