X.
On obtiendra une formule encore plus compliquée,
composée de quinze éléments, en suivant la seconde méthode
générale[1] pour exprimer la nature des surfaces
courbes. Il est cependant très-important de l’élaborer
aussi. En conservant les signes de l’art. IV, posons, en
outre,
![{\displaystyle {\frac {d^{2}x}{dp^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70e6a6ad6487dc41a91cc23dfc4ff999ce15b9ac) |
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![{\displaystyle \qquad {\frac {d^{2}x}{dpdq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9cf511c2e750cf31601bfad0b3440c989580de9) |
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![{\displaystyle {\frac {d^{2}x}{dq^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcf002f57d6236cae6d80355b0fb87d8b0e4787) |
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![{\displaystyle {\frac {d^{2}y}{dp^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/929dfbb3fd1b1832c56ec2b15abd09922dd61ae4) |
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![{\displaystyle \qquad {\frac {d^{2}y}{dpdq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a1333104da2cc0fbfb3d6efb82e0fdea6824ec) |
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![{\displaystyle {\frac {d^{2}y}{dq^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65285af6a13e7fc6cebb1179139228d493eba3a5) |
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![{\displaystyle {\frac {d^{2}z}{dp^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93d0e31880745eebea83272f552d0ef7355e4e88) |
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![{\displaystyle \qquad {\frac {d^{2}z}{dpdq}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8e1301538e6f627cca0cdfdc96328d516d0929) |
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![{\displaystyle {\frac {d^{2}z}{dq^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/962448bff95acff5706d6042d38aecfa98ea970a) |
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Faisons encore, pour abréger,
![{\displaystyle bc'-cb'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f5efa70215b794c324eb3da09349dda7e80035) |
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![{\displaystyle ca'-ac'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d293b6d87672fba1daedeb00eee055742bf7e742) |
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![{\displaystyle ab'-ba'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd78e083f527e4bc3b7eb5c46f0fbab5f85e66a) |
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On a d’abord
![{\displaystyle \mathrm {A} dx\ +\ \mathrm {B} dy\ +\ \mathrm {C} dz\ =\ 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/036990c9ef2369a845c6561bdfcb430ae9821edb)
ou
![{\displaystyle dz\ =\ -{\frac {\mathrm {A} }{\mathrm {C} }}dx\ -\ {\frac {\mathrm {B} }{\mathrm {C} }}dy\ ;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94c34a45a19ca768f151aeecc519c9ff79e73de4)
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 naturellement,
![{\displaystyle \mathrm {C^{3}T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3e2bc7164f245904ca827a93c8bb27e1c0c239) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
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![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36) |
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![{\displaystyle +}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406) |
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![{\displaystyle \mathrm {C^{3}U} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/18935d7969d986bf1c9e01ef925fff3724fa415b) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36) |
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![{\displaystyle +}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406) |
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![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36) |
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![{\displaystyle \mathrm {C^{3}V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d297414569f054d86d774c735d4ec9459ab9643) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
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![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36) |
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![{\displaystyle +}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406) |
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Si donc, pour abréger, nous posons
(1) |
![{\displaystyle \mathrm {A} \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/07bb5804e0e8c153428eda30612df62576952f8d) |
![{\displaystyle +\ \mathrm {B} \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/93856d6ef5a8f1d5a836d43558502f2cc4d84f21) |
![{\displaystyle +\,\mathrm {C} \gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a2db36e10c99982e234945cdc51211733229e9) |
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(2) |
![{\displaystyle \mathrm {A} \alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9ac98657978a05c6a970879c7b928da6326b4a) |
![{\displaystyle +\ \mathrm {B} \beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34352d727aff50a48d3aa8d99d16e1d75bcdd31a) |
![{\displaystyle +\,\mathrm {C} \gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0b2db834c1a61fa5b16e7cdc6a11110432dcb5) |
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(3) |
![{\displaystyle \mathrm {A} \alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea73120a3950648dd5e5bb8cb5c8fa8d66cc8611) |
![{\displaystyle +\,\mathrm {B} \beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6127be2d6ab550e3cff20e6882f3476eb4235406) |
![{\displaystyle +\ \mathrm {C} \gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f1181471897cb94ad5b1a63bbd5eadd712235e1) |
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on a
![{\displaystyle \mathrm {C^{3}T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3e2bc7164f245904ca827a93c8bb27e1c0c239) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
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![{\displaystyle \mathrm {C^{3}U} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/18935d7969d986bf1c9e01ef925fff3724fa415b) |
![{\displaystyle =-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11ee5143dfa298a5d7ce0e739cf0c88852f84a5) |
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![{\displaystyle \mathrm {C^{3}V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d297414569f054d86d774c735d4ec9459ab9643) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
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De là , en faisant le développement,
![{\displaystyle \mathrm {C^{6}(TV-U^{2})} =\mathrm {(DD''-D'^{2})} (ab'-ba')^{2}=\mathrm {(DD''-D'^{2})C^{2},} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7915721fe32d2caf2624d63035572cc70b41a62d)
et, par conséquent, la formule, pour la mesure de la
courbure,
![{\displaystyle k=\mathrm {{\frac {DD''-D'^{2}}{(A^{2}+B^{2}+C^{2})^{2}}}.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a90e08d2dfe211e4bde3816e2a2b57551245c)