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II. A Supplementary Memoir on Caustics. By A. Cayley, F.B.S. 
Received November 15, — Read November 22, 1866. 
It is near the conclusion of my “ Memoir on Caustics,” Philosophical Transactions, 
vol. cxlvii. (1857), pp. 273-312, remarked that for the case of parallel rays refracted at 
a circle, the ordinary construction for the secondary caustic cannot be made use of (the 
entire curve would in fact pass off to an infinite distance), and that the simplest course 
is to measure off the distance GQ from a line through the centre of the refracting circle 
perpendicular to the direction of the incident rays. The particular secondary caustic, 
or orthogonal trajectory of the refracted rays, obtained on the above supposition was 
shown to be a curve of the order 8 ; and it was further shown (by consideration of the 
case wherein the distance GQ is measured off from an arbitrary line perpendicular to 
the incident rays) that the general secondary caustic or orthogonal trajectory of the 
refracted rays was a curve of the same order 8. The last-mentioned curve in the case 
of reflexion, or for [m= — 1, degenerates into a curve of the order 6 ; and I propose in 
the present supplementary memoir to discuss this sextic curve, viz. the sextic curve 
which is the general secondary caustic or orthogonal trajectory of parallel rays reflected 
at a circle. 
1. For parallel rays refracted at a circle, taking the equation of the circle to be 
^ 2 +<y 2 — G and the incident rays to be parallel to the axis of a 1 , then if x—m be an arbi- 
trary line perpendicular to the direction of the incident rays, the secondary caustic is the 
envelope of the circle, 
^ 2 {(#-a) 2 +(y-/3) 2 } — {x— to) 2 =0, 
where (a, /3) are the coordinates of a variable point on the refracting circle, and as such 
satisfy the equation a 2 +/3 2 =l. Or, what is the same thing, writing a= cos 0, /3= sin 0, 
the secondary caustic is the envelope of the circle 
(. — cos 0f-\-{y— sin 4) 2 } — {x— m) 2 =0, 
where 6 is a variable parameter. 
2. The equation may be written 
A cos 20-j-B sin 2^+C cos sin 6 +E=0, 
A=l, 
B=0, 
C=4 j «, 2 ,r— 4 m, 
D=Vy, 
E = - 2^(x’+y'-)-2 ^+ 1 + 2m s , 
where 
