415] ON THE THEORY OF RECIPROCAL SURFACES. 579 
A 
The reciprocal of the first of these is 
o-' = a - n 4- k' - 2/ - 3* - 2 C' - 45' - 3©'; 
viz. writing a = 7i {n — 1) — 26 — 3c, and k = Sn (n — 2) — 66 — 8c, this is 
o-' = 4n (n - 2) - 85 - 11c - 2/ - 3 % ' - 2(7' - 45' - 3©'; 
and it thus appears that the order a' of the spinode curve is reduced by 3 for each 
off-plane to'. 
4. As to the other two equations, writing for p, a their values, these become 
j + 6t 4- 3i + 5/3 4- 6y = b (2n — 4) — 2q, 
2% 4- 3co 4- 4i 4- 13/3 4- 07 = c (on — 12) — 6r 4- 30, 
equations which admit of a geometrical interpretation. In fact, when there is only a 
nodal curve, the first equation is 
j 4- 6t = b (2n — 4) — 2q, 
which we may verify when the nodal curve is a complete intersection, P = 0, Q = 0; 
for if the equation of the surface is (A, B, (7$P, Q) 2 = 0, where the degrees of 
A, B, G, P, Q are n — 2f, n—f—g, n — 2g,f, g respectively, then the pinch-points are 
given by the equations P = 0, Q = 0, AG — B 2 = 0, and the number j of pinch-points 
is thus 
=fg (2n - 2/- 2g), = (2n - 4)fg - 2/g (f+ g - 2) ; 
but for the curve P = 0, Q = 0 we have t = 0, and its order and class are b = fg, 
q—fg (f 4- g — 2), or the formula is thus verified. 
Similarly, when there is only a cuspidal curve, the second equation is 
2% 4- 3«w = c (5n — 12) — 6r 4- 30, 
which may be verified when the cuspidal curve is a complete intersection, P = 0, Q = 0; 
the equation of the surface is here (A, B, CQP, Q) 2 = 0, where AG — B 2 = MP + NQ, 
and the points w are given as the intersections of the curve with the surface 
(A, B, CRN, -Mf = 0. 
Now AG — B- vanishing for P = 0, Q = 0 we must have A = Aa. 2 4- A', B = Aa/3 4- B\ 
C — A/3 2 4- G', where A', B', (7 vanish for P = 0, Q = 0; and thence M=AM' + M", 
N = AN' + N", where M", N" vanish for P=0, Q = 0. The equation 
(A, B, C%N } -My = 0, 
writing therein P = 0, Q = 0, thus becomes A 3 (N'a — JP/3) 2 = 0 ; and its intersections with 
the curve P = 0, Q = 0 are the points P = 0, Q= 0, A = 0 each three times, and the 
points P = 0, Q = 0, N'a — M'a = 0 each twice ; viz. they are the points 2^ 4- 3<w. 
But if the degree of A is = A, then the degrees of N', M', a?, a/3, /3 2 are 2n — 3f—Zg — 
2n — 2f— Sg — X, n — 2f—X, n—f—g — X, n — 2g — X, whence the degree of A 3 (N'a. — M'j3) 
is = on — 6f — 6g, and the number of points is =fg (on — 6f—6g), viz. this is 
=fg (5» - 12) - 6/g (f+ g - 2), 
or it is = c (on — 12)— 6r; so that 0 being =0, the equation is verified. 
73—2