dans le cas de
; or on a, en vertu de l’équation proposée aux différences partielles,
![{\displaystyle {\frac {\partial u}{\partial \alpha }}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \alpha }}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e55c9516d952ba4a9b13570809d1259bd2ea64)
on aura donc
(k)
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|
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En différentiant cette équation par rapport à
on aura
![{\displaystyle {\frac {\partial ^{2}u}{\partial \alpha ^{2}}}={\frac {\partial ^{2}.\int zdu}{\partial \alpha \partial t}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa3af21022b5f580d8e812a471c35c0d41e4fa5)
or l’équation (k) donne, en y changeant
en ![{\displaystyle \int zdu,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e219fe25d90ffa8873b1132f9faa7a5a6cb0b33)
![{\displaystyle {\frac {\partial .\int zdu}{\partial \alpha }}={\frac {\partial .\int z^{2}du}{\partial t}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26912dfbdda6ec7f3f7212bdbc29f2c185729776)
partant,
![{\displaystyle {\frac {\partial ^{2}u}{\partial \alpha ^{2}}}={\frac {\partial ^{2}.\int z^{2}du}{\partial t^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae8f7b9b291f3dd672df2c1a227f7b11696c7db)
En différentiant encore par rapport à
on aura
![{\displaystyle {\frac {\partial ^{3}u}{\partial \alpha ^{3}}}={\frac {\partial ^{3}.\int z^{2}du}{\partial \alpha \partial t^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da58054e198ba8697fe572e88825a035cc9ad8a9)
or l’équation (k) donne, en y changeant
en ![{\displaystyle \int z^{2}du,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45b3e8c6ce67475efa8be6bb0e8db5a2b9cf6342)
![{\displaystyle {\frac {\partial .\int z^{2}du}{\partial \alpha }}={\frac {\partial .\int z^{3}du}{\partial t}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce2110c9d339dd79d8ea20a1981ec099d22308d)
partant,
![{\displaystyle {\frac {\partial ^{3}u}{\partial \alpha ^{3}}}={\frac {\partial ^{3}.\int z^{3}du}{\partial t^{3}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb8f5ee82e65e38373edd98064420b48f2c3355)
En suivant ce procédé, il est aisé d’en conclure généralement
![{\displaystyle {\frac {\partial ^{n}u}{\partial \alpha ^{n}}}={\frac {\partial ^{n}.\int z^{n}du}{\partial t^{n}}}={\frac {\partial ^{n-1}.z^{n}{\frac {\partial u}{\partial t}}}{\partial t^{n-1}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74a05a1d6d2c37f5a75e451d83519bb12c9459a)
Supposons maintenant qu’en faisant
on ait
étant