partant
![{\displaystyle {\begin{aligned}&\mathrm {(RV)^{2}+(VL)^{2}} =\\&\qquad r^{2}\cos ^{2}p+\sin ^{2}\varphi +2\alpha \sin ^{2}\varphi \Gamma (\cos \varphi )-2r\sin \varphi \sin p\sin(q+\varpi )\\&\qquad -2\alpha r\sin \varphi \sin p\sin(q+\varpi )\Gamma (\cos \varphi )+r^{2}\sin ^{2}p\sin ^{2}(q+\varpi )\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b851aedf9d74d5a6ede29f68515f503bedf9cc0c)
mais, le point
étant supposé à la surface du corps, on a, par la nature de la courbe génératrice,
![{\displaystyle \mathrm {(RV)^{2}+(VL)^{2}=\left[1-(CL)^{2}\right]\left[1+2\alpha \Gamma (CL)\right]} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb7553ef1a8fa0181eb6f8613bb71664f1948e39)
Soit
![{\displaystyle \sin ^{2}\varphi \Gamma (\cos \varphi )=\Pi (\cos \varphi )\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9183bb1aad8bd1921a09efebe8744e975b1f86f)
on aura
![{\displaystyle \mathrm {\left[1-(CL)^{2}\right]\left[1+2\alpha \Gamma (CL)\right]=1-(CL)^{2}+2\alpha \Pi (CL)} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef94d37483c6edcff3dfde43dd118e01e1ab7eb1)
donc, en comparant les deux expressions de
et substituant au lieu de
sa valeur, on aura l’équation
![{\displaystyle {\begin{aligned}r-&2r\sin \varphi \sin p\sin(q+\varpi )+2r\cos \varphi \sin p\cos(q+\varpi )\\=&2\alpha \Pi \left[\cos \varphi +r\sin p\cos(q+\varpi )\right]-2\alpha \Pi (\cos \varphi )\\&+{\frac {2\alpha r}{\sin \varphi }}\Pi (\cos \varphi )\sin p\sin(q+\varpi )\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b29c8c3decc53f1d4e2b09c30196435572533909)
d’où l’on tire
![{\displaystyle {\begin{aligned}r=&2\sin \varphi \sin p\sin(q+\varpi )-2\cos \varphi \sin p\cos(q+\varpi )\\&+{\frac {2\alpha }{r}}\Pi \left[\cos \varphi +r\sin p\cos(q+\varpi )\right]-{\frac {2\alpha }{r}}\Pi (\cos \varphi )\\&+{\frac {2\alpha }{\sin \varphi }}\Pi (\cos \varphi )\sin p\sin(q+\varpi ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e756a7e303c5aa6a09ab26aa7fd779577804f032)
Or soit
on aura
![{\displaystyle \mu \sin \varphi =\Pi (\cos \varphi )\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/633b5e9c1802e0abab86f5b2a570b7ed7073eebb)
ensuite
![{\displaystyle \sin \varpi ={\frac {\cos \varphi d\varphi +\alpha d\mu }{\sqrt {d\varphi ^{2}+2\alpha d\mu d\varphi \cos \varphi }}}=\cos \varphi +\alpha {\frac {d\mu }{d\varphi }}\sin ^{2}\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e5e7868ee1eba0197078bec229328d2a06cfaf)
et
![{\displaystyle \cos \varpi ={\frac {\sin \varphi d\varphi }{\sqrt {d\varphi ^{2}+2\alpha d\mu d\varphi \cos \varphi }}}=\sin \varphi -\alpha {\frac {d\mu }{d\varphi }}\sin \varphi \cos \varphi \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c57e276da78e09642aa1ada862e31133a574def)