Page:Annales de mathématiques pures et appliquées, 1827-1828, Tome 18.djvu/63

$x_{1}\operatorname {d} .{\frac {\operatorname {d} x'}{\operatorname {d} s}}+y_{1}\operatorname {d} .{\frac {\operatorname {d} y'}{\operatorname {d} s}}+z_{1}\operatorname {d} .{\frac {\operatorname {d} z'}{\operatorname {d} s}}=0.\qquad$(7)

L’équation (6) établit un relation entre l’arc de la courbe cherchée, ses coordonnées courantes et leurs différentielles ; en la développant, elle prend la forme

$x'^{2}+y'^{2}+z'^{2}-2s\left(x'{\frac {\operatorname {d} x'}{\operatorname {d} s}}+y'{\frac {\operatorname {d} y'}{\operatorname {d} s}}+z'{\frac {\operatorname {d} z'}{\operatorname {d} s}}\right)+s^{2}=a^{2}.\quad$(8)

En intégrant l’équation (7) et remarquant que

${\begin{aligned}\int x_{1}\operatorname {d} .{\frac {\operatorname {d} x'}{\operatorname {d} s}}&=x_{1}{\frac {\operatorname {d} x'}{\operatorname {d} s}}-\int {\frac {\operatorname {d} x'}{\operatorname {d} s}}\operatorname {d} x_{1},\\\\\int y_{1}\operatorname {d} .{\frac {\operatorname {d} y'}{\operatorname {d} s}}&=y_{1}{\frac {\operatorname {d} y'}{\operatorname {d} s}}-\int {\frac {\operatorname {d} y'}{\operatorname {d} s}}\operatorname {d} y_{1},\\\\\int z_{1}\operatorname {d} .{\frac {\operatorname {d} z'}{\operatorname {d} s}}&=z_{1}{\frac {\operatorname {d} z'}{\operatorname {d} s}}-\int {\frac {\operatorname {d} z'}{\operatorname {d} s}}\operatorname {d} z_{1}\end{aligned}}$

on trouve

$x_{1}{\frac {\operatorname {d} x'}{\operatorname {d} s}}+y_{1}{\frac {\operatorname {d} y'}{\operatorname {d} s}}+z_{1}{\frac {\operatorname {d} z'}{\operatorname {d} s}}-\int \left({\frac {\operatorname {d} x'}{\operatorname {d} s}}\operatorname {d} x_{1}+{\frac {\operatorname {d} y'}{\operatorname {d} s}}\operatorname {d} y_{1}+{\frac {\operatorname {d} z'}{\operatorname {d} s}}\operatorname {d} z_{1}\right)=C.$

ou, parce qu’en vertu de l’équation (4) la quantité soumise au signe d’intégration est nulle,

$x_{1}{\frac {\operatorname {d} x'}{\operatorname {d} s}}+y_{1}{\frac {\operatorname {d} y'}{\operatorname {d} s}}+z_{1}{\frac {\operatorname {d} z'}{\operatorname {d} s}}=C.\qquad$(9)

Si l’on désigne par $\varphi$ l’angle que fait la direction du fil avec celle de la normale au point $(x_{1},y_{1},z_{1})$ de la sphère où il se termine, on aura, en vertu d’une formule connue, et parce que