équations entre lesquelles éliminant
on aura d’abord, pour la double équation de la trajectoire décrite
![{\displaystyle \left.{\begin{aligned}&\left\{{\frac {2k{\frac {x-a}{p}}+{\sqrt {4k^{2}\left({\frac {x-a}{p}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\alpha -4k^{2}{\frac {a^{2}}{p^{2}}}}}}{V\operatorname {Cos} .\alpha -2k{\frac {a}{p}}}}\right\}^{p}\\\\=&\left\{{\frac {2k{\frac {y-b}{q}}+{\sqrt {4k^{2}\left({\frac {y-b}{q}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\beta -4k^{2}{\frac {b^{2}}{q^{2}}}}}}{V\operatorname {Cos} .\beta -2k{\frac {b}{q}}}}\right\}^{q}\\\\=&\left\{{\frac {2k{\frac {z-c}{r}}+{\sqrt {4k^{2}\left({\frac {z-c}{r}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\gamma -4k^{2}{\frac {c^{2}}{r^{2}}}}}}{V\operatorname {Cos} .\gamma -2k{\frac {c}{r}}}}\right\}^{r}\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebb96c07ef1f5d88040b7c58feb9b6bb3e41c9d)
(23)
Les mêmes équations (22) peuvent être écrites comme il suit :
![{\displaystyle {\begin{aligned}\left(V\operatorname {Cos} .\alpha -2k{\frac {a}{p}}\right)e^{\frac {2kt}{p}}-2k{\frac {x-a}{p}}&={\sqrt {4k^{2}\left({\frac {x-a}{p}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\alpha -4k^{2}{\frac {a^{2}}{p^{2}}}}},\\\\\left(V\operatorname {Cos} .\beta -2k{\frac {b}{q}}\right)e^{\frac {2kt}{q}}-2k{\frac {y-b}{q}}&={\sqrt {4k^{2}\left({\frac {y-b}{q}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\beta -4k^{2}{\frac {b^{2}}{q^{2}}}}},\\\\\left(V\operatorname {Cos} .\gamma -2k{\frac {c}{r}}\right)e^{\frac {2kt}{r}}-2k{\frac {z-c}{r}}&={\sqrt {4k^{2}\left({\frac {z-c}{r}}\right)^{2}+V^{2}\operatorname {Cos} .^{2}\gamma -4k^{2}{\frac {c^{2}}{r^{2}}}}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28b7c007000fcd6279f4766efd10fa0654bd5f72)
carrant alors les deux membres, réduisant et divisant respectivement par
il viendra, en transposant,