XI.
À l’aide de la formule que nous venons de trouver, nous
en établirons une autre, qui doit être rangée parmi les
théorèmes les plus féconds dans la doctrine des surfaces
courbes. Introduisons les notations suivantes :
![{\displaystyle a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f564e5dc0b6e68af32ca8614e972f5b36e944a24) |
![{\displaystyle +\ b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84402bc81cc2ff16031f238540c6d86efb9de35) |
![{\displaystyle +\ c^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adb3c92d7f09949cf195b245f5a331d039756149) |
|
![{\displaystyle aa'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590bf98afe53a1b33df1a73c9cd2580226efad2f) |
![{\displaystyle +\ bb'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd0cffc5160e50e42a972b0d5d2224403fa1a6d) |
![{\displaystyle +\ cc'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e296d713e7196172935bd9c84fda78fc2fdfe91f) |
|
![{\displaystyle a'^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/989e467cafed73a0c36d39e991d6f12811c5f90a) |
![{\displaystyle +\ b'^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca6419aef7cc4ee9a566c1d7fed04373d4c9151) |
![{\displaystyle +\ c'^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8bd978ec653db850230a1e88795d7e29d455f6) |
|
(4) |
![{\displaystyle a\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b6e4966d1567cf9ecc780858cffb9cb4a9ffe45) |
![{\displaystyle +\ b\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdbc0aa7a2c1ac2d58f82a4743156ccb94119863) |
![{\displaystyle +\ c\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5595c8e32e496d498e8abb2d65c9673ab1621b) |
|
(5) |
![{\displaystyle a\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b93095590870af79643a2791c117515296c9b) |
![{\displaystyle +\ b\beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb01f8ef0e2db620440be4417cd9371e6d8ca1b) |
![{\displaystyle +\ c\gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4f6551853370a3a74e0a7b29413ae85aea31e4) |
|
(6) |
![{\displaystyle a\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b53857622ea3dbc1fb53bd262799f9535db6c5) |
![{\displaystyle +\ b\beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745803019f7b7456e40a63b95893a36c8c9b048c) |
![{\displaystyle +\ c\gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d123b1cfc2d807a06f521b821ed29df905470f) |
|
(7) |
![{\displaystyle a'\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e24182892198fc7930756568a58946f8c5c55f) |
![{\displaystyle +\ b'\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f42d1ec16ba62dc2ab8cf8576a35160220b5ac) |
![{\displaystyle +\ c'\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc0dfafe1420d320c342b74ac240069bf9d6123) |
|
(8) |
![{\displaystyle a'\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f38eea5029b4f6fa2f2710e74e0ae42934e7e95) |
![{\displaystyle +\ b'\beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c7f50eb68de345cae9dad29ffbbe5910c500fc) |
![{\displaystyle +\ c'\gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9910fc51082b9e8a1e9b60026fecc03ab8de7d2e) |
|
(9) |
![{\displaystyle a'\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9caa8fb823b148555a3ac834430c801538e026) |
![{\displaystyle +\ b'\beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf02121b5a45a5b76936f866cae6da317de0b17b) |
![{\displaystyle +\ c'\gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cc505fc2677b1ad41fc9b6e3ea97c3ec9f5093) |
|
![{\displaystyle \mathrm {A^{2}\ +\ B^{2}\ +\ C^{2}\ =\ EG\ -\ F^{2}} \ =\ \Delta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7453b8c02cac1c7b7bffc3fd501d3052fe33e654)
Éliminons des équations (1), (4), (7) les quantités
ce que nous ferons en les multipliant par
,
et les ajoutant, il viendra
![{\displaystyle \qquad \left[A(bc'-cb')+a(b'\mathrm {C} -c'\mathrm {B} )+a'(c\mathrm {B} -b\mathrm {C} )\right]\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/97693ec8d33e2674c58325b959cd1034f6c5b53f)
![{\displaystyle =\mathrm {D} (bc'-cb')+m(b'\mathrm {C} -c'\mathrm {B} )+n(c\mathrm {B} -b\mathrm {C} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa43677f6f27a904d3bf4dbdfd56138b5e8e1c9a)
équation que nous transformons facilement en celle-ci,
![{\displaystyle \mathrm {AD} \ =\ \alpha \Delta \ +\ a(n\mathrm {F} \ -\ m\mathrm {G} )\ +\ \ a'(m\mathrm {F} \ -\ n\mathrm {E} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e04bc7409b2ac0856b2bfebb1afb7357f240554)
De la même manière, l’élimination des quantités
ou
des mêmes équations, donne
![{\displaystyle \mathrm {BD} \ =\ \beta \Delta \ +\ b(n\mathrm {F} \ -\ m\mathrm {G} )\ +\ \ b'(m\mathrm {F} \ -\ n\mathrm {E} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5751466f9976805b6c0605191f2805a23a3ca01b)
![{\displaystyle \mathrm {CD} \ =\ \gamma \Delta \ +\ c(n\mathrm {F} \ -\ m\mathrm {G} )\ +\ \ c'(m\mathrm {F} \ -\ n\mathrm {E} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a76e712f771fa80ab212a44826a9f7179c761cd)
En multipliant ces trois équations par
et les
ajoutant, on obtient
(10) |
|
Si nous traitons de la même manière les équations (2),
(5), (8), il vient
![{\displaystyle \mathrm {AD'} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/684f73e3cc0c158611795bf0e25181391f0226a1) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \alpha '\Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b587872600bdf06678e9448a03cd9b52f69ac4c) |
|
![{\displaystyle \mathrm {BD'} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65446a98635119b97c48d336742990f3a89f256) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta '\Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b283ad08d03fbf0e53418c44acaa64d3ee997bb2) |
|
![{\displaystyle \mathrm {CD'} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8cd3124fdc2a01ac999dd70e299fc9827bd907) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \gamma '\Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/593d96c58f4f076f102d42504f649819e0bdb504) |
|
ces équations étant multipliées par
leur addition donne
![{\displaystyle \mathrm {D'^{2}} =(\alpha '^{2}+\beta '^{2}+\gamma '^{2})\Delta +m'(n'\mathrm {F} -m'\mathrm {G} )+n'(m'\mathrm {F} -n'\mathrm {E} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/489ff80706660ae8f1d617f59255ff1dbb6c1dc2)
La combinaison de cette equation avec l’équation (10)
donne
![{\displaystyle \mathrm {DD''-D'^{2}} =\left(\alpha \alpha ''+\beta \beta ''+\gamma \gamma ''-\alpha '^{2}-\beta '^{2}-\gamma '^{2}\right)\Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9182dd8996da754ec3d1729f47ef30e5dd79c48e)
![{\displaystyle +\mathrm {E} \left(n'^{2}-nn''\right)+\mathrm {F} \left(nm''-2m'n'+mn''\right)+\mathrm {G} \left(m'^{2}-mm''\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e61d7ee414ffcd3da25680fb87851418f7c0c1c8)
Il est évident qu’on a
![{\displaystyle {\frac {d\mathrm {E} }{dp}}=2m,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ee9dcb0bdb73b5cf5521a6b00844b9e264b614) |
![{\displaystyle \;{\frac {d\mathrm {E} }{dq}}=2m',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6817158afc18e63f8df4de8bef587bb7e4ce26) |
![{\displaystyle \;{\frac {d\mathrm {F} }{dp}}=m'+n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3069b38df22acd2898fc21b8751479e64bee5241) |
|
|
![{\displaystyle \;{\frac {d\mathrm {G} }{dp}}=2n',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bf3917c0d9e420439430dc14701521161ef9ed) |
![{\displaystyle \;{\frac {d\mathrm {G} }{dq}}=2n'',\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda114b49ba5bbbfeeb87a3f1389370a59f15f7a) |
|
ou
![{\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) |
![{\displaystyle ={\frac {1}{2}}{\frac {d\mathrm {E} }{dp}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e48d3d6875f928e89db6cbdcef15c23bd9242f7a) |
![{\displaystyle \quad m'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f572ddc68e1ee805e3066e87e6883b9770b4974) |
![{\displaystyle ={\frac {1}{2}}{\frac {d\mathrm {E} }{dq}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c69c9405609b85756a98ff858e117075769d47c8) |
![{\displaystyle m''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87ce1a143d0df33cd77ebfef625b807023c8a5d) |
|
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) |
![{\displaystyle ={\frac {d\mathrm {F} }{dp}}-{\frac {1}{2}}{\frac {d\mathrm {E} }{dq}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b45da7086d774a61aadb02c5284ac01fd94f4067) |
![{\displaystyle \quad n'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36b5c9df51bbe98fdd6342555f61712b12478f65) |
![{\displaystyle ={\frac {1}{2}}{\frac {d\mathrm {G} }{dp}},\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/727ba249da42dace7cbcaa6e8785118036d74f1c) |
![{\displaystyle n''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0cc872b3fe6eb303b8e4a13c26bd7dccc1f7e5) |
|
D’ailleurs on peut facilement s’assurer qu’on a
![{\displaystyle \alpha \alpha ''+\beta \beta ''+\gamma \gamma ''-\alpha '^{2}-\beta '^{2}-\gamma '^{2}={\frac {dn}{dq}}-{\frac {dn'}{dp}}={\frac {dm''}{dp}}-{\frac {dm'}{dq}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14e1380c421cebc68c2c35b6bb667aa3f530b171)
![{\displaystyle =-{\frac {1}{2}}.{\frac {d^{2}\mathrm {E} }{dq^{2}}}+{\frac {d^{2}\mathrm {F} }{dp.dq}}-{\frac {1}{2}}.{\frac {d^{2}\mathrm {G} }{dp^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2196df5247a4f31d12823fed8b4bdfcda6711a1e)
Si nous substituons ces diverses expressions dans la formule que nous avons trouvée à la fin de l’article précédent pour la mesure de la courbure, nous parvenons à la formule suivante, qui ne contient que les seules quantités
et leurs quotients différentiels du premier et du second ordre,
![{\displaystyle 4(\mathrm {EG-F^{2})} ^{2}k=E\left[{\frac {d\mathrm {C} }{dq}}.{\frac {d\mathrm {G} }{dq}}-2{\frac {d\mathrm {F} }{dp}}.{\frac {d\mathrm {G} }{dq}}+\left({\frac {d\mathrm {G} }{dp}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fabe8a29e4853446f5efc6b7de8168dc718cf6f)
![{\displaystyle +\mathrm {F} \left({\frac {d\mathrm {E} }{dp}}.{\frac {d\mathrm {G} }{dq}}+{\frac {d\mathrm {E} }{dq}}.{\frac {d\mathrm {G} }{dp}}-2{\frac {d\mathrm {E} }{dq}}.{\frac {d\mathrm {F} }{dq}}+4{\frac {d\mathrm {F} }{dp}}.{\frac {d\mathrm {F} }{dq}}-2{\frac {d\mathrm {F} }{dp}}.{\frac {d\mathrm {G} }{dp}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f131d789b1fc8b90d31617cf7d4381a53ce925)
![{\displaystyle +G\left[{\frac {d\mathrm {E} }{dp}}.{\frac {d\mathrm {G} }{dp}}-2{\frac {d\mathrm {E} }{dp}}.{\frac {d\mathrm {F} }{dq}}+\left({\frac {d\mathrm {E} }{dq}}\right)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114761ea7de46b7bfec5531c540c7e8f12a8c035)
![{\displaystyle -2(\mathrm {EG-F^{2}} )\left({\frac {d^{2}\mathrm {E} }{dq^{2}}}-2{\frac {d^{2}\mathrm {F} }{dp.dq}}+{\frac {d^{2}\mathrm {G} }{dp^{2}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908e28d88b160c41411b31e2ea53f62a263ce065)