380 
A MEMOIR UPON CAUSTICS. 
[145 
whence replacing q, p by their values, the required equation is 
(1 - /x 2 ) 2 (x 2 + y 2 - l) 2 (/x 2 O 2 + if) - l) 2 
+ 2 {p 2 (x 2 -4- y 2 ) - 2yu 2 + 1) (2/x 2 {pc 2 + y 2 ) - p 2 - 1) (/x 2 O 2 +y 2 ) - 2 + /x 2 ) ¿c 2 - 27/xU 4 = 0, 
which is the equation of an orthogonal trajectory of the refracted rays. 
In the case of reflexion, p = — 1, and the equation becomes 
4 {x 2 + y 2 — l) 3 — 27 x 2 = 0. 
Comparing this with the equation of the caustic, it is easy to see, 
Theorem. In the case of parallel rays and a reflecting circle, there is a secondary 
caustic which is a curve similar to and double the magnitude of the caustic, the 
position of the two curves differing by a right angle. 
XLI. 
The entire system of the orthogonal trajectories of the refracted rays might in 
like manner be determined by finding the envelope of the circle (where, as before, 
a, /5 are variable parameters connected by the equation a 2 + /3 2 = 1 ), 
p 2 {{x — ot) 2 + (y — f3) 2 } — (a + m) 2 = 0. 
{The result, as far as I have worked it out, is as follows, viz.— 
(3 — 12 [m 2 + 2nip 2 x + /x 4 (x 2 + y 2 )~\ + [1 — 2p 2 + 2m 2 — 2p 2 (x 2 + y 2 )] 2 ) 3 
— ([1 — 2p 2 + 2m 2 — 2p 2 (x 2 + y 2 )\ [9 + 18;n 2 + 36mp 2 x + 18/x 4 (x 2 + y 2 )] 
— 54 [to 2 + 2mp 2 x + p 4, {x 2 — y 2 )] — [1 — 2p 2 + 2to 2 — 2p 2 (x 2 + y 2 )] 3 ) 2 = 0, 
which, it is easy to see, is an equation of the order 8 only. Added Sept. 12.—A. C.}