[145 145] A MEMOIR UPON CAUSTICS. 367 
vals, one or what is the same thing, 
rthogonal 
y caustic 
a = a' , 
G = C , 
y=y 
(1) 
or a real 
a' 
a 
a' 
(«) 
;he same 
a = —2, 
y 2 
c= 
=Z>~r 
c y 
a' 
a = -7 a . 
c' 
C =— , 
1 
/* =-/ 
(/3) 
y 2 
/* 
a = a' , 
a' 
c = ~ , 
a' 
^ = ? 
(7) 
iltiplying 
c' 2 
a= a" 
equation 
c — c' , 
c'y 
" = 7 
(«) 
c' 2 
a =—,, 
c' 
C = 
c' 
p = 
00, 
a 
CL 
or what is again the same thing, 
c' 2 c 2 
a' a 
(1) 
a' = a, 
a 
y 2 y 2 
expressed 
/ c 2 
c' 2 a 
a' 
00 
a = - , 
~,= a 
a 
a fj? 
y 2 
, a 
a = - 2 > 
y 2 
c' 2 c 2 
a' a ’ 
a' 
— 2 = a 
y 2 
(/3) 
a' — a , 
c' 2 a 
“~7 o ? 
a' _c 2 
(7) 
a /¿ 2 
/a 8 a 
/ c 8 
a = - , 
c' 2 
-7 = a , 
a' a 
/u/ 2 ~~ /a 2 
(8) 
a 
a 
, a 
c' 2 
a' _c 2 
(0, 
= 0. 
a = — 2 , 
— = a , 
y 2 a 
y 2 
a 
we have in each case identically the same secondary caustic, and therefore also 
identically the same caustic; in other words, the same caustic is produced by six 
different systems of a radiant point and refracting circle. It is proper to remark that if 
we represent the six systems of equations by (a', d, y) = (a, c, y), (a', c', y) = a (a, c, y), 
&c., then, a, /3, y, 8, e will be functional symbols satisfying the conditions 
1 = oc/3 = /3a = y 2 = 8 2 = e 2 , 
a = /3 2 = 8y = e8 = ye, 
/3 = a? = y 8 = 8e = ey, 
y = 8a = cue = e/3 = /38, 
8 = ea = a y = y/3 = /3e, 
6 = y a= a8 = 8/3 = /3y.