Et comme
![{\displaystyle \mathrm {P} '={\frac {d\mathrm {P} }{dz}},\quad \mathrm {Q} '={\frac {d\mathrm {Q} }{dz}},\quad \mathrm {R} '={\frac {d\mathrm {R} }{dz}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d761a180ced932da9bbaf5f52c2e3a6ea3f661e2)
on aura, par la différentiation,
![{\displaystyle {\begin{aligned}\mathrm {PP'} =&{\frac {\sin ^{2}\omega \sin 2z}{2}},\\\mathrm {QQ'} =&\mathrm {RR'} =-{\frac {\sin ^{2}\omega \sin 2z}{4}},\\\mathrm {P^{2}P'} =&{\frac {\sin ^{3}\omega (\cos z-\cos 3z)}{4}},\\\mathrm {Q^{2}P'} +&\mathrm {2PQQ'} =\mathrm {R^{2}P'+2PRR'} \\&\qquad \qquad \qquad ={\frac {\left(4-3\sin ^{2}\omega \right)\sin \omega \cos z+3\sin ^{3}\omega \cos 3z}{8}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ff0e68db155611111f663ff07e9268867ef5ee)
toutes les autres quantités
étant nulles.
Faisant donc ces substitutions dans les formules du no 19, et supposant, pour abréger,
![{\displaystyle {\begin{aligned}\mathrm {B} =&\mathrm {M} \left(a^{2}-{\frac {b^{2}+c^{2}}{2}}\right)\left(1-{\frac {3\sin ^{2}\omega }{2}}\right),\\\mathrm {C} =&{\frac {3\mathrm {M} }{2}}\left(a^{2}-{\frac {b^{2}+c^{2}}{2}}\right)\sin ^{2}\omega ,\\\mathrm {D} =&{\frac {3\mathrm {M} }{2}}\left(2f^{3}-3f'^{3}-3f''^{3}\right)\left(\sin \omega -{\frac {5}{4}}\sin ^{3}\omega \right),\\\mathrm {E} =&{\frac {5\mathrm {M} }{8}}\left(2f^{3}-3f'^{3}-3f''^{3}\right)\sin ^{3}\omega ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da8766c7c775f58f9f1c08ac9d11cb6c990ffdb)
on aura, à cause de la non-sphéricité de la Terre
Force perturbatrice dans la direction du rayon,
![{\displaystyle -{\frac {3}{2}}{\frac {\mathrm {B+C} \cos 2z}{x^{4}}}-2{\frac {\mathrm {D} \sin z+\mathrm {E} \sin 3z}{x^{5}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/746b4562d947149d35c39557cfc4948556e19a35)
Force perturbatrice perpendiculaire au rayon,
![{\displaystyle {\frac {\mathrm {C} \sin 2z}{x^{4}}}-{\frac {\mathrm {D} \cos z+3\mathrm {E} \cos 3z}{2x^{5}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9e8df4602e127da291a7ae524d3a8f93e142bb)