BIE. A. CAYLEY’S lilEMOIE UPON CAUSTICS. 
297 
XXVI. 
Suppose next that rays proceeding from a point are refrmted at a circle. Take the 
centre of the circle as origin, let the radius be c, and take rj as the coordinates of the 
radiant point, a, j3 the coordinates of the point of incidence, cc, y the coordinates of a point 
in the refracted ray : then the general equation 
VQGN'h-/!^' W V^GN"=0 
becomes, taking the centre of the circle as the point N on the normal, or writing a=:0, &=(), 
- {(a:—ay+(y—l3y}((3a:—u}jy+y>m^—oiy-\-(}j—(3f}((^j'—ayfz=0; 
or putting and expanding, 
{2{ti^X—(Jtj^y^^)} 
d-a"/3 { —i(^)]X—iJj^xy^)-\-2{)j^y—iMy7j)} 
{—^{lriy—yxy7i)-\-2{l^x—ycifl)) 
{2{fy-yxy] 
—a" {(^+/+c*y— 
+ 2a/3 { +y^ + (f)lri—y{%^ + ??' + c')xy } 
— /S' + — jW;'(f + J7' + C'>'} 
= 0 , 
which may be represented by 
Aa'+Ba'|3+Ca(3'+D/3'+Fa'+Gai3+H/3'=0. 
Now a'+j3'=c', and we may write 
The equation thus becomes 
+'f(^+?) (--1) =0 ; 
or expanding. 
(A-Bi-C-Dz> 
+?(F-Gi-H>’ 
3 
4-(3A-B*+C+3D^)z 
+!(F+H) 
+ (3A+Bi+C-3Di)j 
+?(F+Gi-H)i 
+(A+B/ — C+D*)^ 
2 R 
MDCCCLVII. 
