12 
PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 
The corresponding values of x are 
#=cos0(2 cos 2 0— 2mcos0)+m, —m— cos0, m 2 — 2) ; 
each of the points in question, viz. the points 
x—\{m^-^m 2 -- 2), y= 0, 
is a node on the axis of x. 
12. It is to be observed that for m<\/2 the nodes are both imaginary; for m—y / 2 
they coincide together at the point x=—j= ; for m>y/ 2 they are both real : it is to be 
v 2 
further noticed that 
node x=^(m+y/m 2 —2) corresponds to cos 0=|(m— \/ m 2 — 2), 
where (m being >\/2) the point (cos 0, sin 0) is a real point on the circle x 2 -\-y 2 =l ; 
in fact for m < § (that is, m=y / 2 to m=f) we have \{m— y/m 2 — 2) <%m, that is, 
cos0<f; but m = or >-§, then cos 0=i(m— y/rn 2 — 2)= 7=5== is = or <•§, and 
J ' m+ vm *— 2 
node x=^(m—\/m 2 — 2) corresponds to cos 0=^(m+\/m 2 — 2), 
where (m being >\/2) the point (cos0, sin0) is a real point on the circle x 2 -\-y 2 = 1 so 
long as m is not >f, that is, from m=\/ 2 to m—\\ but if m> f, then the point in 
question is an imaginary point on the circle — whence also the node x —\{m — y/ m 2 — 2) 
is an acnode or isolated point. 
In the case m=| we have 
node x=l corresponding to cos0=^ or 0=60°, 
„ x=\ „ cos 0=1 or 0=0°, 
the last-mentioned point x—\ being in fact the point of union of two cusps in the case 
m= § now in question. Hence in this case we have at (x=^, «/=0) a triple point equi- 
valent to two cusps and a node ; visibly, there is only a single branch cutting the axis 
of x at right angles. 
In the case m=\/ 2, the nodes coincide as above mentioned at the point x=-^r= on 
the axis ; for this value of m the coordinates of the cusps are 
a--HV2 (=ff which is<^) ; V=±tt- 
13. Starting from the equation 1024 (x— m) 2 U=S 3 — T 2 =0, it is clear that the cusps 
are included among the intersections of the curves S=0, T=0; these two curves intersect 
in 24 points which lie 9 + 9 at the circular points at infinity, 2 + 2 at the points x=m, 
y 1 — 1=0, and 1+1 are the cusps, or points x=m—^-,y 2 =(l—^-^ • Toverifythis, 
writing for a moment 
S' = (r*+y 2 — m 2 — 1) 2 + 6m(x—m), 
T=z2(x 2 +y 2 —m 2 —l) 3 + lSm(x—m)(x 2 -\-y 2 —m 2 —l)—27(x—m) 2 , 
