139
ET DES SURFACES COURBES.
![{\displaystyle {\frac {(x'+x)^{2}}{a^{2}}}+{\frac {(y'+y)^{2}}{b^{2}}}+{\frac {(z'+z)^{2}}{c^{2}}}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de16e647957b2c8807317caefd405155592dd360)
![{\displaystyle {\frac {(x'+x)^{2}}{a'^{2}}}+{\frac {(y'+y)^{2}}{b'^{2}}}+{\frac {(z'+z)^{2}}{c'^{2}}}=1\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e048152164cf9c04cb715f43ba6847dd1871e7)
puis, en développant ;
![{\displaystyle 0=\left({\frac {x'^{2}}{a^{2}}}+{\frac {y'^{2}}{b^{2}}}+{\frac {z'^{2}}{c^{2}}}-1\right)+{\frac {2x'x}{a^{2}}}+{\frac {2y'y}{b^{2}}}+{\frac {z'z}{c^{2}}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4d0d8d129a824cf59c89a4432d24cf8baeeffe)
![{\displaystyle 0=\left({\frac {x'^{2}}{a'^{2}}}+{\frac {y'^{2}}{b'^{2}}}+{\frac {z'^{2}}{c'^{2}}}-1\right)+{\frac {2x'x}{a'^{2}}}+{\frac {2y'y}{b'^{2}}}+{\frac {z'z}{c'^{2}}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb28cb823ac14c8a8afd66d3ff277c5b42cbc6d)
Parce que le point
est sur la courbe, nous aurons
d’abord les deux équations de condition.
![{\displaystyle {\frac {x'^{2}}{a^{2}}}+{\frac {y'^{2}}{b^{2}}}+{\frac {z'^{2}}{c^{2}}}=1,\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/39da0190a936d4cfa32be6351de7ecdea9aff0c6)
(5)
![{\displaystyle {\frac {x'^{2}}{a'^{2}}}+{\frac {y'^{2}}{b'^{2}}}+{\frac {z'^{2}}{c'^{2}}}=1\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8dd99adc67d93f6f10225d40d7730b57054ee48)
(5′)
et les équations de la tangente, rapportée aux axes primitifs
seront
![{\displaystyle {\frac {x'(x-x')}{a^{2}}}+{\frac {y'(y-y')}{b^{2}}}+{\frac {z'(z-z')}{c^{2}}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f28ce44c363a98c77ef611e778776830cd0a8e35)
![{\displaystyle {\frac {x'(x-x')}{a'^{2}}}+{\frac {y'(y-y')}{b'^{2}}}+{\frac {z'(z-z')}{c'^{2}}}=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25a2789d71e06b66e07e84b6d44cc0504d92ab7a)
ou simplement, en vertu des conditions (5, 5′)
![{\displaystyle {\frac {x'x}{a^{2}}}+{\frac {y'y}{b^{2}}}+{\frac {z'z}{c^{2}}}=1.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/50982e5cf149fce97555b63e9f2679fb259e64fc)
(6)
![{\displaystyle {\frac {x'x}{a'^{2}}}+{\frac {y'y}{b'^{2}}}+{\frac {z'z}{c'^{2}}}=1.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0c1cba7eef11f284e966009c58797469b4ebcc)
(6′)