The Collected Mathematical Papers of Arthur Cayley volume 8
The Collected Mathematical Papers of Arthur Cayley volume 8
Arthur Cayley
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J ecris aussi Abe b c, B ca c a, C = ab a b, E = a 2 + a 2 + a" 2, F= aa f + bb + cc, G = a"> +b 2 + c 2 . Liquation differentielle des courbes de courbure est dx, dy, dz = 0. A, B, C dA, dB, dC premier terme de ce determinant est dx (B dC - C dB), savoir : (adh + a dk) { B [(a/3 - ba. + V* - a /3) dh + (#" - ba" + b a - aft ) dk] - C [(ca - ay + a y - c a) dh + (ca" - ay" + a y - c a. ) dk]}, C. VIII. 34 266 SUR LES SURFACES DIVISIBLES EN CARRES PAR LEURS COURSES [517 ce qui se r^duit tout de ...suite a (adh + a dk} { [a (Aa + 5/3 + Cy ) - a (Aa + 5/3 + Cy )] dh - [a (Aa" + 5/3" + Cy") - a (Aa + 5/3 + Cy )] dk] ; en formant les expressions analogues du second et du troisieme terme, et en prenant la somme, liquation devient [E(Aa f + 5/3 + Cy )-F (Aa + 5/3 + Cy) ] dh* + [E (Aa" + B/3" + Cy") -G(Aa + B/3 +Cy)]dh dk + [F(Aa" + B$" + Cy") - G (Aa + 5/3 + #/)] d& = 0, ou, ce qui est la meme chose, dk\ -dhdk, dh z = E, F, G la+B@+Cy, Aa + 5/3 + Cy, A a" + Bft" + Cy" celle-ci est liquation diffe rentielle des courbes de courbure d une surface quand les coordonne es x, y, z d un point de la surface sont donnees comme fonctions de deux para- metres h, k.
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