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Poincaré - Théorie des tourbillons, 1893.djvu/41
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33
ÉQUATION GÉNÉRALE DE CES SURFACES
Je dis que :
d
ψ
d
α
=
d
T
d
α
.
{\displaystyle {\frac {d\psi }{d\alpha }}={\frac {d\mathrm {T} }{d\alpha }}.}
En effet
d
ψ
d
α
=
∂
ψ
∂
x
ξ
+
∂
ψ
∂
y
η
+
∂
ψ
∂
z
ζ
.
∂
ψ
∂
x
=
d
u
d
t
=
u
∂
u
∂
x
+
v
∂
u
∂
y
+
w
∂
u
∂
z
,
etc.
{\displaystyle {\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 :
d
ψ
d
α
=
u
(
ξ
∂
u
∂
x
+
η
∂
v
∂
x
+
ζ
∂
w
∂
x
)
+
v
(
ξ
∂
u
∂
y
+
η
∂
v
∂
y
+
ζ
∂
w
∂
y
)
+
w
(
ξ
∂
u
∂
z
+
η
∂
v
∂
z
+
ζ
∂
w
∂
z
)
.
{\displaystyle {\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 :
d
T
d
α
=
u
d
u
d
α
+
v
d
v
d
α
+
w
d
w
d
α
=
u
(
ξ
∂
u
∂
x
+
η
∂
v
∂
x
+
ζ
∂
w
∂
x
)
+
v
(
ξ
∂
u
∂
y
+
η
∂
v
∂
y
+
ζ
∂
w
∂
y
)
+
w
(
ξ
∂
u
∂
z
+
η
∂
v
∂
z
+
ζ
∂
w
∂
z
)
.
{\displaystyle {\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 :
ξ
∂
u
∂
x
+
η
∂
u
∂
y
+
ζ
∂
u
∂
z
=
ξ
∂
u
∂
x
+
η
∂
v
∂
x
+
ζ
∂
w
∂
x
.
…
etc.
{\displaystyle {\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}}}