02 FHESNEL, OX DOU13LE KEFKACTION. 



arrived at by following another method. An equation of the 

 first degree in (o*) is easily obtained by causing the cutting plane, 

 and therefore the tangent plane which is parallel to it, to vary, 

 so that [dv) may be nothing; then the common intersection of 

 the two successive positions of the tangent plane is the tangent 

 which passes through the foot of the perpendicular dropped from 

 the origin of coordinates on the tangent plane ; and this tangent 

 passing through the point of contact, may serve to determine its 

 position as well as the tangent plane, and by the same method 

 of differentiation and elimination. 



If we diffei'entiate equation (A.), considering (o) as constant, 

 we find 



dn _ m {b^ — o'^) 

 dm n{a^ — u'^) ' 



Differentiating in the same way the equation (B.) of the tangent 

 plane, we have 



dn _ u^m + X {z — mx — ny) 

 dm u^n + y {z — mx — ny)' 



Equating these two values, we get the relation 



[v-n + y {z — inx — ny)^ {b^ — w^) m 



= [u^ m + X [z — m X — ny)'] (o^ — u^) n, 

 in which the two terms containing (u*) destroy each other, and 

 which becomes 



m n [a^ — b^) v^ + [z — m x — ny) {my — n x) ii^ 

 + {z — mx — ny) {nax'^ — m by'^) = 0; 



i "^ ~~' Vfi OC — ft t/t 



or, putting for u'^ its value ^^^- ^ 2 j ^^^ suppressing the 



common factor {z — mx — ny), 

 {z — mx — ny)^ {my — n x) + mn {a^ — h^) {z — mx — ny) 



+ {na^x - m b^y) (1 + tn^ + n^) = (C.) 



Now, to obtain the surface of the wave, it is sufficient to differ- 

 entiate this equation successively with respect to (m) and {n), 

 and afterwards to eliminate (m) and (n) by aid of these two new 

 equations. 



Having found the equation of the surface of the wave by a 

 much shorter process, it was sufficient for me to verify it by its 



dz 

 satisfying equation (C), in which {m) and {n) represent the -r- 



and -T- of the surface sought. I have followed this synthetical 



