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 ;