A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
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[359 
or writing (x, y) in place of (X, F), that is taking now (x, y) as the current 
coordinates of a point in the tangent, the equation of the tangent is 
x cos 26 — y sin 20 — 2 cos 0 + m = 0; 
whence observing that this equation may be expressed as a rational equation of the 
fourth order in terms of the parameter tan \0 (or cos 0 + V - i sin 6), it appears that 
the class of the secondary caustic is = 4. 
8. The secondary caustic may be considered as the envelope of the tangent, and 
the equation be obtained in this manner. Comparing with the general equation 
we have 
and thence 
giving 
if for a moment 
A cos 20 + D sin 20 + G cos 0 + D sin 0 + E = 0, 
A = x, 
B = -y, 
c =- 2, 
I)= 0, 
E = m, 
S = 4 {3 (x 2 + y 2 ) + m 2 — 3}, 
T = 4 {18m {oc? + y 2 ) — 27x — 2m 3 + 9m}, 
£ 3 -T 2 = 16F, 
V= 4 {3 (x 2 + y 2 ) + m 2 — 3} 3 — {18m (cc 2 + y 2 ) — 27x — 2m 3 + 9m} 2 . 
The equation of the curve is thus obtained in the form V= 0; this should of 
course be equivalent to the before-mentioned equation U = 0; and by developing V, 
and comparing with the second of the two expressions of U, it appears that we in 
fact have V = 27 U. 
9. Taking as parameter tan \0, or if we please cos 0 + V — 1 sin 0, the foregoing 
values of (x, y) in terms of 0 give (x, y, 1) proportional to rational and integral 
functions of the degree 6 in the parameter; so that not only the curve is a sextic 
curve, but it is a unicursal sextic, or curve of the order 6 with the maximum number, 
= 10, of nodes and cusps; that is, if 8 be the number of nodes and k the number 
of cusps, we have 8 + k = 10. Moreover, introducing the same parameter into the 
equation of the tangent, this equation is seen to be of the degree 4 in the 
parameter; that is, the class of the curve is = 4: this implies 23 + 3/c = 26, and we 
have therefore 3 = 4, tc = 6. To verify these numbers, it is to be remarked that it 
appears by the equation of the curve that there is at each of the circular points at 
infinity a triple point in the nature of the point x = 0, y = 0 on the curve y 3 =x*;