378 
A MEMOIR UPON CAUSTICS. 
[145 
XXXIX. 
Again, to compare the general equation with that previously obtained for parallel 
rays refracted at a circle, we must write y — j n , c = 1, a = so, Z = k (for the equation 
of the refracted ray was taken to be Xx + Yy + k = 0); we have then 
and, after the substitution, a= oo. The equation becomes in the first instance 
k 6 + 2k 3 X |i (l + (1+ k 2 ) a 2 ) k 2 - %k 2 aX 2 | + (X 2 + F 2 ) (l + (1 + k 2 ) a 2 ) k 2 - \khiX 2 1' 
- B Y 2 (1 + k 2 + k 2 a 2 + 2k 2 aX) = 0 ; 
and then putting a = so, or, what is the same thing, attending only to the terms 
which involve a 2 , and throwing out the constant factor we obtain 
(.X 2 + F 2 ) (X 2 - 1 - k 2 ) 2 - 4k 2 Y 2 = 0, 
or 
X 2 (X 2 -1 -k 2 ) 2 + Y 2 (X + 1 +k)(X- 1 -k)(X + l-k) (X- 1 -k) = 0 
which agrees with the former result. 
XL. 
It was remarked that the ordinary construction for the secondary caustic could 
not he applied to the case of parallel rays (the entire curve would in fact pass off 
to an infinite distance), and that the simplest course was to measure the distance 
GQ from a line through the centre of the refracting circle perpendicular to the 
direction of the rays. To find the equation of the resulting curve, take the centre of 
the circle as the origin and the direction of the incident rays for the axis of x; let 
the radius of the circle be taken equal to unity, and let y denote, as before, the 
index of refraction. Then if a, ¡3 are the coordinates of the point of incidence of a 
ray, we have a 2 + [3 2 — 1, and considering a, /3 as variable parameters connected by this 
equation, the required curve is the envelope of the circle, 
y 2 {(x — a) 2 + (y - ¡3) 2 } — a 2 = 0. 
Write now a = cos 0, ¡3 = sin 6, then multiplying the equation by — 2, and writing 
1 + cos 20 instead of 2 cos 2 0, the equation becomes 
1 + cos 20 — 2y 2 (x 2 + y 2 — 2x cos 0 — 2y sin 0 + 1) = 0, 
which is of the form 
A cos 20 + B sin 20 + G cos 0 + D sin 0 + E = 0,