528 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT 



, , , 0/ ''-'^1 , ^/ dF^ , „dd) , 



Hence, uda- + vdy + ivdx •= -~. —— da; + ^.-- — - dv + f —T-dz 

 ^ F, d!tj F. dy ^ ■' dz 



= Fr' f)" \{F. ^' d^ + F, '^-p dy) <P + F,f/-^ dz] 

 ' dx dy ^ dz ' 



i-i 1-1 

 = /^/ F,' .d.F^F,(p. 



Consequently the required factor is Z', '' ■ F-^ ' ; and the differential equation of the surface 

 cutting at right angles the directions of the motion, is d-F^F^cp = 0. If \^ = 0, be the equation 

 of this surface, we have -^ = F^F.;,<p + a function of/. We may now proceed to find a value 

 of -r + r^ , the sum of the reciprocals of the principal radii of curvature of the surfiice at any 



point, by substituting in the general expression fo"" "h + ^> '^'^•' 



' fd'^ d"\|/ d-y\,\ ld\\r' d\l/ d^'\ d\j/ d\\,- rf'x/, rfx//' 



\dx'' dy' dz' j Kdar dy' dx^ I d,v' dx' dy dy' 



d'yj/ d\\r" d''-^ d\j/ d\|/ d^'^ d\j/ d\l/ d'\jy d\p d\j/ 



dz- dz'' dwdy doe dy dxdz dx dz dydz dy dz 



j df dx//' dv//l 





Now '^^f!'" .F.!'' .u, and -^ = f'' '' . F,'' ' .v; and therefore -^^ = o if u = o, and 

 dx ' dy dx 



r = if u = 0. As we shall require the value oi — + -p only for points where u = and 

 dy K K 



d^ d\|/ 



V = 0, we shall suppose in the general expression that ^ = " and — = 0. Hence 



1 1 dx-' dy' ^ - dx' ^ 



R'*' R' d^l, F F.'^ 



dm ' ' dz 



dy' 



Taking now the equation (3) obtained in page 365 of the Paper on Luminous Rays, and sup- 

 posing it to apply to any point of the plane perpendicular to the axis of sr in which («= and « = 0, 

 we shall have, neglecting small terms. 



~dt- 



if- ' dz- ' dz ' \R R'J 



That this equation may be of the form --^ - c"". -^- = 0, which, for the reasons given 

 in the Paper just cited, it is required to be, we must have 



' • ~d^ U «'/ ^ ' dz^ 



^c'-^k^f, \i c'" = c'"- (\ + k). 

 dz" 



