24 



Prof. Cayley on the 



[Nov. 16, 



viz. writing a=»(» — 1) — 25 — 3c, and n=3?i(n — 2) — 65 — 8c, this is 



a' = 4»(n - 2) - 85 - 1 1 c - 2/ ' - 3 X ' - 2C' - 4 B' - 3 w'j 



and it thus appears that the order a of the spinode curve is reduced by 3 

 for each off-plane w'. 



4. As to the other two equations, writing for p, a their values, these 

 become 



j+ 67 + 3i + 5/3 + 6y = h(2n - 4) - 2q, 

 2 x +3o> + 4« + 18/3 + 5 y =c(5»- 12)— 6r+30, 



equations which admit of a geometrical interpretation. In fact when there 

 is only a nodal curve, the first equation is 



which we may verify when the nodal curve is a complete intersection, P=0, 

 Q— ; for if the equation of the surface is (A, B, C^P, Q) 2 =0, where 

 the degrees of A, B, C, P, Q are n — 2f 3 n—f—g, n — 2y,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-2 9 )=(2n-4Vg-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 



2 X + 3w = c(5n - 1 2) - 6r + 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, Cf£P, Q) 2 = 0, 

 where AC — B 2 =MP+NQ, and the points x , w are given as the intersec- 

 tions of the curve with the surface (A, B, C^N, — M) 2 =0. 



Now AC— B 2 vanishing for P=0, Q = we must have A=Aa 2 +A', 

 B = Aa/3 + B', C = Aj3 2 + C, where A', B', C 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 X N , — M) 2 =0, writing therein P=0, 

 Q = 0, thus becomes A 3 (N f a — M'/3) 2 = ; 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 x + 3w. 



But if the degree of A is =X, then the degrees of N', M', a 2 , a/3, /3 2 are 

 2n-3f-2g-\, 2n~2f-3g-\ n-2f—X, n—f—g—X, n ~2g—\ 

 whence the degree of A 3 (.N'a — M'/3) is — 5n—6f—6g, and the number of 

 points is =fg(5n — 6f—6g) J viz. this is =/<7(5ra— 12)— 6fg(f+g — 2), or 

 it is =c(5n— 12) — 6r ; so that 6 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 x should be — G x instead of + 6^, 



