124] ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE CIRCLE. 121 
or, again, 
r=£ , 
c' 2 C 2 
r = r 
II 
MX 
(1) 
* £ ’ 
c' 2 
G =p 
g £» 
r =f 
fl 2 
(«) 
II 
c' 2 c 2 
r = I’ 
fl' 2 ? ’ 
08) 
r=f > 
c' 2 f 
f ~y ’ 
1' c 2 
t,' 2 ~r 
(7) 
£' = 
* 1 ’ 
c' 2 
— =P 
gr £ > 
II 
(*) 
l'=i 
b 0 > 
/J, 2 
r' 2 
H ’ 
r c 2 
^ % ’ 
(«) 
then, whichever system of values of c', /jl be substituted for f, c, /1, we have in 
each case identically the same secondary caustic, the effect of the substitution being 
simply to interchange the different forms of the equation; and we have therefore 
identically the same caustic. By writing * 
(f> O', fl')= (f, C, fl) 
= a (£ c, fi), 
&c., 
a, /3, y, S, e will be functional symbols, such as are treated of in my paper “ On the 
Theory of Groups as depending on the symbolic equation 6 n = 1,” [125], and it is 
easy to verify the equations 
1 = a/3 = /3a = 7 2 = 8 2 = e 2 , 
a = /3 2 = By = e8 = ye, 
/3 = a 2 = ey =7 8 = Be, 
y = 8a — e/3 = ¡38= ae, 
8 = ea = y/3 = ay — {3e, 
e = yet. = 8/3 = /37 = a§. 
Suppose, for example, %= — c, i.e. let the radiant point be in the circumference; 
then in the fourth system P = - c, c' = - -, (or, since c' is the radius of a circle, this 
radius may be taken -), n' = — 1, or the new system is a reflecting system. This is 
f 1 
one of M. St Laurent’s theorems, viz. 
C. II. 
16