118 
[124 
124. 
ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE 
CIRCLE. 
[From the Philosophical Magazine, vol. vi. (1853), pp. 427—431.] 
M. St Laurent has shown (Gergonne, vol. xvm. [1827] p. 1), that in certain cases 
the caustic by refraction of a circle is identical with the caustic of reflexion of a circle 
(the reflecting circle and radiant point being, of course, properly chosen), and a very 
elegant demonstration of M. St Laurent’s theorems is given by M. Gergonne in the 
same volume, p. 48. A similar method may be employed to demonstrate the more 
general theorem, that the same caustic by refraction of a circle may be considered as 
arising from six different systems of a radiant point, circle, and index of refraction. 
The demonstration is obtained by means of the secondary caustic, which is (as is well 
known) an oval of Descartes. Such oval has three foci, any one of which may be 
taken for the radiant point: whichever be selected, there can always be found two 
corresponding circles and indices of refraction. The demonstration is as follows:— 
Let c be the radius of the refracting circle, /a the index of refraction; and taking 
the centre of the circle as origin, let £, y be the coordinates of the radiant point, 
the secondary caustic is the envelope of the circle 
A( x - a + y — ft) — (£ — a + 7) — ¡3~) = Q, 
where cl, /3 are parameters which vary subject to the condition 
a 2 + /3 2 — c 2 = 0; 
the equation of the variable circle may be written 
{/a 2 (x 2 + y 2 + c 2 ) — (p + y 2 + c 2 )} — 2 {y?x — %) cl — 2 (g 2 y — y) /3 = 0,