PROFESSOR CAYLEY ON THE THEORY OF RECIPROCAL SURFACES. 
85 
4. As to the other two equations, writing for g, <r their values, these become 
y+6£+3£+5/3-{-6y=i(2w— 4)— 2q, 
2%-j- 3<y -f- M + 18/3 + 5y =c(5n— 12)— 6r+ 30, 
equations which admit of a geometrical interpretation. In fact, when there is only a 
nodal curve, the first equation is 
j -j- Qt=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, CJ/T, Q) 2 = 0, where the degrees of A, B, C, 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, AC — B 2 — 0, and the number^” of pinch-points is thus 
=fg(2n-2f- 2g)=(2n - A)fg-2fg(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-\-g— < 2‘), or the formula is thus verified. 
Similarly, when there is only a cuspidal curve, the second equation is 
-%+3<v=c(5'«— 12) — 6r + 35, 
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, C^P, Q) 2 =0, where AC — B 2 =MP+NQ, 
and the points %, a are given as the intersections of the curve with the surface 
(A, B, C£N, — M) 2 =0. 
Now AC — B 2 vanishing for P — 0, Q = Q we must have A = Aa 2 + A', B = Aa0-f-B', 
C=A3 2 +C / , where A', B', C' vanish for P = 0, Q = 0; and thence M = AM'+M ,/ , 
N = AN , d-N ,/ , where M", N" vanish for P=0, Q = 0. The equation 
(A, B, CXN, — M) 2 =0, 
writing therein P = 0, Q=0, thus becomes A 3 (N'a — M'3) 2 =0 ; and its intersections with 
the curve P = 0, Q=Q 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^+3<y. 
But if the degree of A is =\, then the degrees of N', M', a 2 , a/3, 3 2 are 2n—2>f—2g — \ 
2n — 2f—og—\ n~2f—\ n—f—g — X, n—2g — 'k, whence the degree of A 3 (N'a — M'3) 
is =5n — 6f— Qg, and the number of points is =fg(5n—6f—Gg), viz. this is 
=fg{5n-l2)-6fg(f+g - 2), 
or it is =c(5n— 12)— 6r ; so that 0 being =0, the equation is verified. 
5. It was also pointed out to me by Dr. Zeuthen that in the value of 24t given in 
No. 10 the term involving % should be —6% instead of +6^, and that in consequence 
the coefficients of ^ are erroneous in several others of the formulae. Correcting these. 
