68 Proceedings of the Royal Irish Academy. 



point (.'-•„, //„, z ) is also varied, the total variation of E between the two points 

 may be written shortly 



BE = \ji(ISx tm$y+ »Sz)]' 







1 



= S[u ' - c) + my 4 



o 



-[jL\{x-c)M + y$m + n$z\], 







where c lias the (constant) values c , c, at the two ends of the path. If 

 then, 



i 

 If" = [E - fi\l(x - c) { iii ;i - /c!] , 







since 



/' + !»' + /('• = 1, III I IKCIH r HCIl = 0, 



it follows thai 



8JF = - [ M |y-^(x-c))8m f M (2-/ I ' -']'• 



' '0 



tation ■'!" this expression in terms of t he intersecti E the 



ray and plane 



X V -ij Z - : 



I in ii 



is immediate, and shows that 



<■'"'-» - < > • fi I ' ' ■ I . . . <1 



when now the coordinates of the intersections of the ray 



with thi A A" W is the value of E between the feet of the 



perpendiculars from ind c, 0, n the ray, and is a minimum in the 



■ . - /. 



2. Let the axis ol * be the axis of a symmetrical optical system, the 



:tion of the i ; thai of i ring. This direction will In- called 



•• t" the right" Let the effect of refraction at a single surface be considered. 



The refractive index of the medium to the left will 1"- denoted by fu. x , of that 



the right by « . The radius of curvature of the surface being r,- at the 

 vertex, the coordinates of the vertex are taken to , 0), and those of 



the fcurvatun 0, 0), so that n is negative when the ray m» 



a concave surface. The surfs evolution, its equation may be 



written 



A 1 V /, 1 Y /, I +-&,)+,.., 



where 6, = for a spherical surface and b, = - 1 for a parabolic surface, thus 



