Page:Lagrange - Œuvres (1867) vol. 1.djvu/676

2^o On aura

$s=\mathrm {\frac {I+J}{D}} {\sqrt {1+q^{2}}}+\eta \cos(\varphi -\omega ),$

d’où l’on tire

${\frac {ds}{d\varphi }}=\mathrm {\frac {I+J}{D}} {\frac {qdq}{d\varphi {\sqrt {1+q^{2}}}}}-\eta \sin(\varphi -\omega )+{\frac {d\eta }{d\varphi }}\cos(\varphi -\omega )+{\frac {d\omega }{d\varphi }}\eta \sin(\varphi -\omega ).$

Supposons ici, à l’imitation de ce que nous venons de faire plus haut,

$\cos(\varphi -\omega )d\eta =-\eta \sin(\varphi -\omega )d\omega ,$

de manière que l’on ait simplement

${\frac {ds}{d\varphi }}={\frac {I+J}{D}}{\frac {qdq}{d\varphi {\sqrt {1+q^{2}}}}}-\eta \sin(\varphi -\omega ),$

c’est-à-dire que la variation instantanée du rayon $r={\frac {1}{s}}$ soit la même que si l’ellipse demeurait constante, et différentiant cette valeur de ${\frac {ds}{d\varphi }},$ on trouvera

${\begin{aligned}{\frac {d^{2}s}{d\varphi ^{2}}}=&\mathrm {\frac {I+J}{D}} \left[{\frac {dq^{2}+qd^{2}q}{d\varphi ^{2}{\sqrt {1+q^{2}}}}}-{\frac {q^{2}dq^{2}}{d\varphi ^{2}\left(1+q^{2}\right)^{\frac {3}{2}}}}\right]\\&-\eta \cos(\varphi -\omega )-{\frac {d\eta }{d\varphi }}\sin(\varphi -\omega )+{\frac {d\omega }{d\varphi }}\eta \cos(\varphi -\omega )\,;\end{aligned}}$

or, à cause de ${\frac {dq^{2}}{d\varphi ^{2}}}+q^{2}=\varepsilon ^{2},$

${\frac {dq^{2}}{d\varphi ^{2}{\sqrt {1+q^{2}}}}}-{\frac {q^{2}dq^{2}}{d\varphi ^{2}\left(1+q^{2}\right)^{\frac {3}{2}}}}={\frac {dq^{2}}{d\varphi ^{2}\left(1+q^{2}\right)^{\frac {3}{2}}}}={\frac {\varepsilon ^{2}-q^{2}}{\left(1+q^{2}\right)^{\frac {3}{2}}}}$

$={\frac {1+\varepsilon ^{2}}{\left(1+q^{2}\right)^{\frac {3}{2}}}}-{\frac {1}{\sqrt {1+q^{2}}}}\,;$

de plus

${\frac {d^{2}q}{d\varphi ^{2}}}=-q-{\frac {V}{\mathrm {C} -2\int \mathrm {Q} r^{2}d\varphi }},$

donc

${\begin{aligned}{\frac {qd^{2}q}{d\varphi ^{2}{\sqrt {1+q^{2}}}}}=&-{\frac {q^{2}}{\sqrt {1+q^{2}}}}-{\frac {q}{\sqrt {1+q^{2}}}}{\frac {V}{\mathrm {C} -2\int \mathrm {Q} r^{2}d\varphi }}\\=&-{\sqrt {1+q^{2}}}+{\frac {1}{\sqrt {1+q^{2}}}}-{\frac {q}{\sqrt {1+q^{2}}}}{\frac {V}{\mathrm {C} -2\int \mathrm {Q} r^{2}d\varphi }}\,;\end{aligned}}$