Si l’on substitue pour 
leurs valeurs 
on aura
![{\displaystyle {\begin{aligned}&(q-q')yy'+(p'-p)x'y={\frac {q'-q}{2}}rr'\left[\cos(v'+v)-\cos(v'-v)\right]\\&\qquad \qquad \qquad \qquad \qquad +{\frac {p'-p}{2}}rr'\left[\sin(v'+v)-\sin(v'-v)\right],\\\\&(p-p')xx'+(q'-q)xy'={\frac {p'-p}{2}}rr'\left[\cos(v'+v)+\cos(v'-v)\right]\\&\qquad \qquad \qquad \qquad \qquad +{\frac {q'-q}{2}}rr'\left[\sin(v'+v)+\sin(v'-v)\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/740c22ddf7f2dd282d6a6127840c807f0c28f936)
En négligeant les excentricités et les inclinaisons des orbites, on a
ce qui donne
![{\displaystyle {\frac {1}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{\frac {3}{2}}}}-{\frac {1}{\left[(x'-x)^{2}+(y'-y)^{2}+(z'-z)^{2}\right]^{\frac {3}{2}}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/889520415c1f806f478e82862351e9bc49898e8f)
![{\displaystyle {\frac {1}{a'^{3}}}-{\frac {1}{\left[a^{2}-2aa'\cos(n't-nt+\varepsilon '-\varepsilon )+a'^{2}\right]^{\frac {3}{2}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0b52426503b46230bc44d2ab1c587bc8003e58)
on a de plus, par le no 48,
![{\displaystyle {\frac {1}{\left[a^{2}-2aa'\cos(n't-nt+\varepsilon '-\varepsilon )+a'^{2}\right]^{\frac {3}{2}}}}={\frac {1}{2}}\Sigma {\rm {B}}^{(i)}\cos i(n't-nt+\varepsilon '-\varepsilon ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81507d44304ea4ce1cbf8c843c4be22358701213)
le signe intégral 
s’étendant à toutes les valeurs entières, positives et négatives, de 
en y comprenant la valeur 
; on aura ainsi, en négligeant les termes de l’ordre des carrés et des produits des excentricités et des inclinaisons des orbites,
![{\displaystyle {\begin{aligned}{\frac {dp}{dt}}&={\frac {q'-q}{2c}}{\frac {m'a}{a'^{2}}}\left[\cos(n't+nt+\varepsilon '+\varepsilon )-\cos(n't-nt+\varepsilon '-\varepsilon )\right]\\\\&+{\frac {p'-p}{2c}}{\frac {m'a}{a'^{2}}}\left[\sin(n't+nt+\varepsilon '+\varepsilon )-\sin(n't-nt+\varepsilon '-\varepsilon )\right]\\\\&+{\frac {q'-q}{4c}}m'aa'\Sigma {\rm {B}}^{(i)}\left\{\cos \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )\right]\right.\\&\qquad \qquad \qquad \qquad \left.-\cos \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon \right]\right\}\\\\&+{\frac {p'-p}{4c}}m'aa'\Sigma {\rm {B}}^{(i)}\left\{\sin \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )\right]\right.\\&\qquad \qquad \qquad \qquad \left.-\sin \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon \right]\right\},\\\\{\frac {dq}{dt}}&={\frac {p'-p}{2c}}{\frac {m'a}{a'^{2}}}\left[\cos(n't+nt+\varepsilon '+\varepsilon )+\cos(n't-nt+\varepsilon '-\varepsilon )\right]\\\\&+{\frac {q'-q}{2c}}{\frac {m'a}{a'^{2}}}\left[\sin(n't+nt+\varepsilon '+\varepsilon )+\sin(n't-nt+\varepsilon '-\varepsilon )\right]\\\\&+{\frac {p'-p}{4c}}m'aa'\Sigma {\rm {B}}^{(i)}\left\{\cos \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )\right]\right.\\&\qquad \qquad \qquad \qquad \left.+\cos \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon \right]\right\}\\\\&+{\frac {q'-q}{4c}}m'aa'\Sigma {\rm {B}}^{(i)}\left\{\sin \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )\right]\right.\\&\qquad \qquad \qquad \qquad \left.+\sin \left[(i+1)(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon \right]\right\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07474cf6312b64d01baa9e448d05b110d15ebd04)
