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mais aussi, quand
est considérée comme fonction de
on a
![{\displaystyle {\frac {dz}{dx}}=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/013f97f9d0a1903f55c8c8bad3ddedde636ff168) |
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![{\displaystyle {\frac {dz}{dy}}=u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc4c4a5b5dca184ecde25ba1c7cdebd2df797f1) |
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Nous tirons de
![{\displaystyle \mathrm {C} dp=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/980076bde9b3691b94c70e15efcf1e14f296aeb6) |
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![{\displaystyle \mathrm {C} dq=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/599229a0d007eeaa71ced3f83f9e13ceaf31f8cd) |
![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36) |
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Les différentielles complètes de
et de
sont donc
![{\displaystyle \mathrm {C} ^{3}dt=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820fe39df57b05cf27c6c8bfb3f3f4c83609b65e) |
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![{\displaystyle \mathrm {C} ^{3}du=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14f591551f7f52cc71363bbed91b72a88d1df14) |
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Si maintenant, dans ces formules, nous substituons
![{\displaystyle {\frac {d\mathrm {A} }{dp}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59d0f3d3b2709a6e5cb1cc84321531cb4aabf71) |
![{\displaystyle c'\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73be6e83ab1f51fe5cee3c2a96cc12998f580ff6) |
![{\displaystyle +\ b\gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/756aef7601558a532491b2d379967c429b9d0e73) |
![{\displaystyle -\ c\beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ca6f8ee5346e4fade4657d9ba456197596a681) |
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![{\displaystyle {\frac {d\mathrm {A} }{dq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a14f1165593a16ffbf8021085b1f3049a8f26f) |
![{\displaystyle c'\beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d868276990f9783bbf9d325370d668c183b02dfe) |
![{\displaystyle +\ b\gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38f5663f236ed5a401f40cd97f4d64561ae2b9f1) |
![{\displaystyle -\ c\beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9148b4dbd9d09d4bad369ee48fbfe125bd25c7ed) |
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![{\displaystyle {\frac {d\mathrm {B} }{dp}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2edfed664f061e128364f8f5f137e4e162a4efde) |
![{\displaystyle a'\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17a3643eb1fdb3fba2e989fbb8a7c02bfe5b0c6) |
![{\displaystyle +\ c\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b91251ef6b1ad6c85cacb3738a6512149accb76f) |
![{\displaystyle -\ a\gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab852c5fbc348178a4b8c68d635104cd5731376) |
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![{\displaystyle {\frac {d\mathrm {B} }{dq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01261b0b8d8fec37b72aa398c21a3250717b0656) |
![{\displaystyle a'\gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a3dabab7968341002ae1ec014cec8d6cdbd7fa) |
![{\displaystyle +\ c\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84b011424931904603699e0d980f27439bb20961) |
![{\displaystyle -\ a\gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7bb2447e145e171051c405d7f9fad81d234fab) |
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![{\displaystyle {\frac {d\mathrm {C} }{dp}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82dd3c83691c255b2519fe3d7308b6bcc091df29) |
![{\displaystyle b'\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a17afdf5986092376dea7f0883eedde0d6cc05d4) |
![{\displaystyle +\ a\beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/017c260f5256a5090e50630b7535d6df1c3dc87f) |
![{\displaystyle -\ b\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0307f0344b315cd5f4c10f38130d6b7b219f1e45) |
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![{\displaystyle {\frac {d\mathrm {C} }{dq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c6cb586538114c9b82f18ddaef3fe074ba056a) |
![{\displaystyle b'\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f537288d2a9d2863037f00056af46f0c8a0152b) |
![{\displaystyle +\ a\beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78e874c3c12b112661647ab818077af384895754) |
![{\displaystyle -\ b\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/668a5f5f51798e6f9e4aa2c1859d20a1847b0334) |
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et si nous considérons que les valeurs des différentielles
ainsi obtenues, doivent être respectivement égales, indépendamment des différentielles
aux quantités
nous trouverons, après quelques transformations qui se présentent assez