454 
[359 
359. 
A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clvii. (for the 
year 1867), pp. 7—16. 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, [145], 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 p = — 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 
sc 2 + y 2 — 1, and the incident rays to be parallel to the axis of x, then if x = m be an 
arbitrary line perpendicular to the direction of the incident rays, the secondary caustic 
is the envelope of the circle 
p 2 {{x — of + (yy — /3) 2 } — (x — nif = 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$, /3 = sin0, 
the secondary caustic is the envelope of the circle 
p 2 {{x — cos 6) 2 + (y— sin 6) 2 ) ~{x — m) 2 = 0, 
where 6 is a variable parameter.