300 
ME. A. CAYLEY’S iMEMOIE TPOY CALSTICS. 
or what is the same thing, 
or what is again the same thing, 
a—d. 
c~d. 
y.= 
7 
( 1 ) 
a’ 
d 
d 
(«) 
C=-,5 
F 
f/.= 
dd 
a! 
Oj — 
d 
c=-r 
F 
y= 
1 
f' 
m 
a=.d. 
d 
F'- 
d 
d 
(7) 
a=-,-> 
a' 
€=c', 
dd 
V 
(^) 
c '2 
d 
d 
/ \ 
a=—i 
a' 
C=-,5 
F 
'd 
(O5 
ing, 
d=a, 
II 
d 
f'^- 
a 
7 
( 1 ) 
a 
d 
(«) 
« =-5 
a 
II 
\l 
a 
1 « 
II 
d 
7^- 
■a 
(/ 3 ) 
d^_a 
d 
d 
(7) 
d=a, 
d fj?' 
a 
d^ 
d 
a 
(^) 
d=—> 
a 
—,=^ai 
d 
f'^~ 
7 
, a 
d^ 
d 
d 
( 2)5 
a =-25 
1 ^ 
- 7 =«: 
d 
”^2~ 
F 
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 («', c', c, (ju), (a', c', (Jb')=c6{a, c, yj), See., then a, (3, y, $ 
will be functional symbols satisfying the conditions 
1 =aj3=|8c6=y^ =s^ 
a=j3^=By=2§ =ys 
I3=k^ =£y 
Y—^ci =ae = 2/3 =(3^ 
d = 2 a =ay=y/3 = (32 
2 =ya =^/3 =|8y. 
XXIX. 
The preceding formulse, which were first given by me in the Philosophical Magazine, 
December 1853, include as particular cases a preceding theorem wnth respect to the 
