on aura
![{\displaystyle \mathrm {T} =2a^{\frac {3}{2}}\left({\text{ϐ}}-e\sin {\text{ϐ}}\cos {\text{ϐ}}'\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317597e14ac4625e46051acd4b874f73fac20de6)
Si l’on ajoute l’une à l’autre les deux expressions de
et de
en
et
et si l’on observe que
![{\displaystyle \cos u'+\cos u=2\cos {\text{ϐ}}\cos {\text{ϐ}}',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7f405f4ae467d267ca33693d089e8e99205949)
on aura
![{\displaystyle R=2a(1-e\cos {\text{ϐ}}\cos {\text{ϐ}}').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08850434d5d448adf274fca6b287bc9e5779f266)
Maintenant soit
la corde de l’arc elliptique ; on a
![{\displaystyle c^{2}=r^{2}+r'^{2}-2rr'\cos(v-v')\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f966f5a7eb8aa422221be2f4489b3eccce6177d)
mais les deux équations
![{\displaystyle r={\frac {a\left(1-e^{2}\right)}{1+e\cos v}},\qquad r=a(1-e\cos u)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7601219bac7ac401891998f4575424284d4f0e4d)
donnent celles-ci
![{\displaystyle \cos v=a{\frac {\cos u-e}{r}},\qquad \sin v={\frac {a{\sqrt {1-e^{2}}}\sin u}{r}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a49773381b75d70020ed29ff9633c4de25c19b7f)
On a pareillement
![{\displaystyle \cos v'=a{\frac {\cos u'-e}{r'}},\qquad \sin v'={\frac {a{\sqrt {1-e^{2}}}\sin u'}{r'}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c90576f2cd625018da89dfa674c2ba8e784c4d9)
on aura donc
![{\displaystyle rr'\cos(v-v')=a^{2}(e-\cos u)(e-\cos u')+a^{2}\left(1-e^{2}\right)\sin u\sin u',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3ec798ff7c7fb5145eafb6f1c7b8efdaf72fe9)
et par conséquent
![{\displaystyle c^{2}=2a^{2}\left(1-e^{2}\right)(1-\sin u\sin u'-\cos u\cos u')+a^{2}e^{2}(\cos u-\cos u')^{2}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6da84ad5bf8854e3fe3b8246f5320d99d66ff6e)
or on a
![{\displaystyle \sin u\sin u'+\cos u\cos u'=2\cos ^{2}{\text{ϐ}}-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bef635a1514f39ed8e71e5edc22ad28725cc71)
![{\displaystyle \cos u-\cos u'=2\sin {\text{ϐ}}\sin {\text{ϐ}}'\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fff3edc55f82e18566048bfc050ac180e4f7bc)
partant
![{\displaystyle c^{2}=4a^{2}\sin ^{2}{\text{ϐ}}\left(1-e^{2}\cos ^{2}{\text{ϐ}}'\right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5b0f38f4a04e188ee0070d200e4f81a6bc2e2e)