PEOFESSOE CAYLEY’S SUPPLEMENTAEY MEMOIE ON CAUSTICS. 
13 
then we have 
T'— 2(.r 2 +^ 5! — m 2 — l)S'=6m(#— m)(^ 2 +^ 2 — m 2 — 1)— 27 (#— mf 
=3(x— m){2m(x 2 +y*— m?— 1)— 9(x— m)}; 
so that the equations S=0, T=0, or, what is the same thing, S'=0, T'=0 give 
(x— w){2m(# 2 -f/— m 2 — 1)— 9(x— m)}=0, 
9 
that is, x— m=0, or else of+y 1 — to 2 — m ). And combining herewith the 
equation S'=(^ 2 +^ 2 — to 2 — 1) 2 +6to(#— to)= 0, we have m=0, — 1) 2 ^0, or else 
81 
(af+f—ni 2 — 1 f=4^2 (^r— m) 2 =6m(^— m), 
and therefore 
(x-m)^{27(x-m)—8m 3 } = 0, 
the second factor of which gives x—m — -^ 7 -to 3 , and thence ;r 2 -f-y 2 — to 2 — 1== — |~to 2 , that 
is, # 2 -l-?/ 2 =l — ^m 2 , and therefore y 2 =(l— fw 2 )— (w^-TyTO 3 ) 2 , ==(1 — |to 2 ) 3 , that is, we 
have 
x=m—i T m\ f=(l-%m 2 )\ 
which, as appears above, gives the two cusps. 
14. Similarly, in the equation 16V— S 3 — T 2 ;^=Q, the intersections of the curves 
S=0, T=0 must include the cusps; the curves in question are the two circles 
3 (^ ? +y 2 )-pm 2 — 3=0, 
1 8m(x 2 -Vy 2 ) — 2 7x' — 2 m 3 + 9 to = 0, 
meeting in the circular points at infinity, and in the two cusps. It is to be added that 
the tangent at the cusp coincides with the tangent of the last-mentioned circle, 
18m(x*+y 2 )—Z7x—2m 3 -\-9m==Q, 
or, as this may also be written, 
15. The axis of x meets the secondary caustic in the two nodes counting as 4 inter- 
sections, and besides in 2 points, viz. the points x=2—m, x— — 2— to ; these correspond 
to the values $ = 0 and $=t respectively. But to verify them by means of the equation 
16V=S 3 — T 2 =0 
of the curve, it may be remarked that for y=0 we have 
S=4(3# 2 +to 2 - 3), T = 4(1 8mx 2 —27x—2m 3 + 9m) ; 
and writing herein x=^r2-~m, we find 
S=4(2to+3) 2 , T=8(2to+ 3) 3 , 
values which satisfy the equation S 3 — T 2 =0. 
