145] 
A MEMOIR UPON CAUSTICS. 
359 
is greater, equal, or less than or to y = 2a 2 Vl —a 2 , according as a > = or < 
depends on the identity (4a 2 — 1) — 16a 6 (1 — a 2 ) = (2a 2 — l) 3 (2a 2 +1). 
1 
vl’ 
this 
To find the points of intersection with the reflecting circle, x 2 +y 2 —1 = 0, we have 
(3a 2 - 1 - 2ax) 2 - 27a 2 (1 - x 2 ) (1 - a 2 ) 2 = 0 ; 
or, reducing, 
8a 3 P + (— 27a 4 + 18a 2 — 15) a 2 x 2 + (54a 4 — 36a 2 + 6) ax + (— 27a 4 + 18a 2 -f 1) = 0, 
i. e. {ax — l) 2 (8ax — 27a 4 + 18a 2 +1) = 0. 
The factor (ax — l) 2 equated to zero shows that the caustic touches the circle in 
the points x = -, y = ± a /1 — ^, i. e. in the points in which the circle is met by the 
CL V CL“* 
polar of the radiant point, and which are real or imaginary according as a > or <1. 
The other factor gives 
27a 4 - 18a 2 -1 
x = - - . 
8 a 
Putting this value equal to +1, the resulting equation is (a + 1) (27a 2 + 9a 4- 1) = 0, and 
it follows that x will be in absolute magnitude greater or less than 1, i.e. the points 
in question will be imaginary or real, according as a>l or a< 1. 
It is easy to see that the curve passes through the circular points at infinity, 
and that these points are cusps on the curve; the two points of intersection with the 
axis of x are cusps (the axis of x being the tangent), and the two points of inter 
section with the ’ circle x 2 + y 2 — a 2 — 0 are also cusps, the tangent at each of the cusps 
coinciding with the tangent of the circle; there are consequently in all 6 cusps. 
XXII. 
To investigate the position of the double points we may proceed as follows: write 
for shortness P = (4a 2 — 1) {x 2 + y 2 ) — 2ax - a 2 , Q = ayS, S = x+y 2 — a 2 ; the equation of the 
caustic is 
hence, at a double point, 
P 3 — 27Q 2 = 0 ; 
18 
ax 
0, 
P 2 ^-18Q^ = 0; 
ay dy 
one of which equations may be replaced by 
dP clQ_ dP dQ_ 0 
dx dy dy dx