Page:Poincaré - Théorie des tourbillons, 1893.djvu/41

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ÉQUATION GÉNÉRALE DE CES SURFACES

Je dis que :

${\frac {d\psi }{d\alpha }}={\frac {d\mathrm {T} }{d\alpha }}.$

En effet

${\begin{aligned}&{\frac {d\psi }{d\alpha }}={\frac {\partial \psi }{\partial x}}\xi +{\frac {\partial \psi }{\partial y}}\eta +{\frac {\partial \psi }{\partial z}}\zeta .\\&{\frac {\partial \psi }{\partial x}}={\frac {du}{dt}}=u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}},\ {\textrm {etc.}}\end{aligned}}$

Substituons :

${\begin{aligned}{\frac {d\psi }{d\alpha }}&=u\left(\xi {\frac {\partial u}{\partial x}}+\eta {\frac {\partial v}{\partial x}}+\zeta {\frac {\partial w}{\partial x}}\right)\\&+v\left(\xi {\frac {\partial u}{\partial y}}+\eta {\frac {\partial v}{\partial y}}+\zeta {\frac {\partial w}{\partial y}}\right)\\&+w\left(\xi {\frac {\partial u}{\partial z}}+\eta {\frac {\partial v}{\partial z}}+\zeta {\frac {\partial w}{\partial z}}\right).\end{aligned}}$

D'autre part :

${\begin{aligned}{\frac {d\mathrm {T} }{d\alpha }}&=u{\frac {du}{d\alpha }}+v{\frac {dv}{d\alpha }}+w{\frac {dw}{d\alpha }}\\&=u\left(\xi {\frac {\partial u}{\partial x}}+\eta {\frac {\partial v}{\partial x}}+\zeta {\frac {\partial w}{\partial x}}\right)\\&+v\left(\xi {\frac {\partial u}{\partial y}}+\eta {\frac {\partial v}{\partial y}}+\zeta {\frac {\partial w}{\partial y}}\right)\\&+w\left(\xi {\frac {\partial u}{\partial z}}+\eta {\frac {\partial v}{\partial z}}+\zeta {\frac {\partial w}{\partial z}}\right).\end{aligned}}$

Mais nous avons vu [18] que :

${\begin{aligned}\xi {\frac {\partial u}{\partial x}}+\eta {\frac {\partial u}{\partial y}}+\zeta {\frac {\partial u}{\partial z}}=\xi {\frac {\partial u}{\partial x}}+\eta {\frac {\partial v}{\partial x}}+\zeta {\frac {\partial w}{\partial x}}.\\\ldots {\textrm {etc.}}\end{aligned}}$