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EXTENSION DE LA MÉTHODE DE M. BOHLIN.
Avec notre nouvelle notation, l’équation (8 bis) doit donc s’écrire
![{\displaystyle \sum {\frac {d\Theta }{dz_{i}}}{\frac {\partial \mathrm {T} _{1}}{\partial u_{i}}}=\Phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b724023e7f36b82c53a7b80986cf74ba741460d)
D’autre part, on a identiquement
![{\displaystyle \theta =\Theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4f1f5722185540d904a05b9cafc9436353d173)
et, comme
ne dépend que des ![{\displaystyle z_{i}^{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5c71f78d9249ae3c52ee9c98442c89c6ab1ea0)
![{\displaystyle {\frac {\partial \Theta }{\partial u_{i}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ab831dd4199eea14db4affd44495bb891b47e8)
Cette équation peut encore s’écrire
![{\displaystyle {\frac {d\Theta }{du_{i}}}+\sum {\frac {d\Theta }{dz_{k}}}{\frac {\partial z_{k}}{\partial u_{i}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48b5d9c401cbba02db6e6a4b6599a9338eb1a38a)
On a d’ailleurs
![{\displaystyle {\frac {\partial \mathrm {T} _{1}}{\partial u_{i}}}={\frac {d\mathrm {T} _{1}}{du_{i}}}+\sum {\frac {d\mathrm {T} _{1}}{dz_{k}}}{\frac {\partial z_{k}}{\partial u_{i}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44ba147c3d2dd69dcb0ba9014d633ef911e3f60)
On trouve alors successivement en transformant (8 bis)
![{\displaystyle \sum {\frac {d\Theta }{dz_{i}}}{\frac {d\mathrm {T} _{1}}{du_{i}}}+\sum {\frac {d\Theta }{dz_{i}}}{\frac {d\mathrm {T} _{1}}{dz_{k}}}{\frac {\partial z_{k}}{\partial u_{i}}}=\Phi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ff56ce9149fb7cf1c1a5a6137278cbfb81f267)
ou, par permutation d’indices,
![{\displaystyle \sum {\frac {d\Theta }{dz_{i}}}{\frac {d\mathrm {T} _{1}}{du_{i}}}+\sum {\frac {d\Theta }{dz_{k}}}{\frac {d\mathrm {T} _{1}}{dz_{i}}}{\frac {\partial z_{k}}{\partial u_{i}}}=\Phi \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e98b0d1d784a26617697a2121547638508119b43)
car
![{\displaystyle {\frac {\partial z_{k}}{\partial u_{i}}}={\frac {\partial z_{i}}{\partial u_{k}}}={\frac {\partial ^{2}\mathrm {T} _{0}}{\partial u_{i}\,\partial u_{k}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab81c5521406260564c35be258a2ece03df0da1)
d’où
![{\displaystyle \sum \left({\frac {d\Theta }{dz_{i}}}{\frac {d\mathrm {T} _{1}}{du_{i}}}-{\frac {d\Theta }{du_{i}}}{\frac {d\mathrm {T} _{1}}{dz_{i}}}\right)=\Phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70d78128559cfa6990ae0d4b03ef144e1cbdab04)
ou, en prenant pour variables les
et les ![{\displaystyle z_{i}^{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5c71f78d9249ae3c52ee9c98442c89c6ab1ea0)
![{\displaystyle \sum \left({\frac {d\Theta }{dz_{i}^{0}}}{\frac {d\mathrm {T} _{1}}{du_{i}^{0}}}-{\frac {d\Theta }{du_{i}^{0}}}{\frac {d\mathrm {T} _{1}}{dz_{i}^{0}}}\right)=\Phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aecca2e0cfbb13c867a4494a678847916ca6813)
Comme
se réduit à
qui ne dépend pas des
il vient enfin
(8 ter)
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