364 
A MEMOIR UPON CAUSTICS. 
[145 
XXVI. 
Suppose next that rays proceeding from a point are refracted at a circle. Take 
the centre of the circle as origin, let the radius be c, and take f, y as the coordi- 
nates of the radiant point, a, /3 the coordinates of the point of incidence, x, y the 
coordinates of a point in the refracted ray: then the general equation 
.-~qG 2 V QGN* + f QG" VqGN* = 0 
becomes, taking the centre of the circle as the point N on the normal, or writing 
a = 6, 6 = 0, 
- {(x - a) 2 + (y - fi) 2 } (/3£ - a??) 2 + /¿ 2 {(£ - a) 2 + (y - fi) 2 } (fix - ay) 2 = 0 ; 
or putting a 2 + fi 2 — c 2 , and expanding, 
a 3 {2 (y 2 x - /¿yf)} 
+ a 2 fi {- 4 (fyx - y?xy%) + 2 (yhy - y?y 2 y)} 
+ a/3 2 {- 4 (fry - y?xyy) + 2 (i?x - fx 2 %)} 
+ fi 3 {2 (ify - y 2 x 2 y)} 
— a 2 [(x 2 + y 2 + c 2 ) 7] 2 — ¡i 2 (f 2 + rj 2 + c 2 ) y 2 } 
+ 2a/3 {(x 2 + y 2 + c 2 ) %y — y 2 (f 2 -f y 2 + c 2 ) xy) 
— fi 2 {(x 2 + y 2 + c 2 ) f 2 — y? (I 2 + y 2 + 0 2 ) X 2 } 
= 0, 
which may be represented by 
Aa 3 + Ba 2 fi + Gafi 2 + Dfi 3 + Fa 2 + Gafi + Hfi- = 0. 
Now a 2 + /3 2 =c 2 , and we may write 
The equation thus becomes 
or expanding, 
(A — Bi — C — Di) z 3 
+ -(F-Gi-H) z 2 
+ (3ol — Bi + C + 3Di) z 
+ -AF+H) 
1 
+ (3 A + Bi + G — 3Di) - 
+ -(F+Gi-H) I 
c z 2 
+ (A+Bi-C + Di) \ 
z 3 ;