Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/79

partant,

$\cos \theta ={\frac {\sin \theta }{2\theta }}+{\frac {\partial z}{\partial \varpi }},$

ce qui donne

${\frac {\partial z}{\partial \varpi }}=\cos \theta -{\frac {\sin \theta }{2\theta }}\,;$

on a donc ainsi, à l’origine de l’intégrale,

$z=\sin \theta ,\qquad {\frac {\partial z}{\partial \theta }}={\frac {\sin \theta }{2\theta }},\qquad {\frac {\partial z}{\partial \varpi }}=\cos \theta -{\frac {\sin \theta }{2\theta }}.$

On a ensuite (art. IX), dans le cas présent,

$z^{(1)}={\frac {\partial z}{\partial \theta }}-{\frac {z}{\varpi +\theta }}\qquad {\text{et}}\qquad z^{(2)}={\frac {\partial z^{(1)}}{\partial \theta }}+{\frac {z^{(1)}}{\varpi +\theta }},$

d’où l’on tire, à l’origine de l’intégrale,

$z^{(1)}=0\qquad {\text{et}}\qquad z^{(2)}={\frac {\partial z^{(1)}}{\partial \theta }}\,;$

or on a

${\frac {\partial z^{(1)}}{\partial \theta }}={\frac {\partial ^{2}z}{\partial \theta ^{2}}}-{\frac {\cfrac {\partial z}{\partial \theta }}{\varpi +\theta }}+{\frac {z}{(\varpi +\theta )^{2}}},$

ce qui donne, à l’origine de l’intégrale,

${\frac {\partial z^{(1)}}{\partial \theta }}={\frac {\partial ^{2}z}{\partial \theta ^{2}}}\,;$

ainsi, pour avoir ${\frac {\partial z^{(1)}}{\partial \theta }}$ et, par conséquent, $z^{(2)}$ à l’origine de l’intégrale, il ne s’agit que de connaître à ce point la valeur de ${\frac {\partial ^{2}z}{\partial \theta ^{2}}}.$ Pour cela, on observera que

$d{\frac {\partial z}{\partial \theta }}={\frac {\partial ^{2}z}{\partial \theta ^{2}}}\partial \theta +{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}\partial \varpi ={\frac {\partial ^{2}z}{\partial \theta ^{2}}}\partial \theta +{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}\partial \theta \,;$

or on a

$d{\frac {\partial z}{\partial \theta }}=d{\frac {\sin \theta }{2\theta }}={\frac {\cos \theta d\theta }{2\theta }}-{\frac {\sin \theta d\theta }{2\theta ^{2}}},$

partant,

${\frac {\cos \theta }{2\theta }}-{\frac {\sin \theta }{2\theta ^{2}}}={\frac {\partial ^{2}z}{\partial \theta ^{2}}}+{\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}\,;$