1897-98.] Mr E. J. Nanson on a Problem of Sylvester s. 157 
15. Cayley’s first paper at once suggests a generalisation of 
Sylvester’s elimination problem. Denoting by A, B, C, D, F, G, 
H, L, M, N the co -factors of a , b , c, d, f, g, h, l, m, n in the 
determinant 
a h g l x 
h b f m y 
g f c n z 
l m n d w 
x y z w . 
it may be proposed to eliminate x , y , z , w from any independent 
set of four of the ten equations 
A = 0, B = 0 , . . . N = 0 ... (8) 
Now if the conicoid 
(a, b, c, d,f , g, h, l , m, n) (£, rj, l w) 2 = 0 . . (9) 
be a cone, and 
(x, y, z, w) (£, rj, o)) = 0 . . . (10) 
be a proper tangent plane thereto, the ten equations (8) are obviously 
all satisfied. For these equations are found by expressing that 
the line of intersection of (10) with an arbitrary plane touches the 
conicoid (9). 
Hence it may be inferred, first, that the discriminant of (9) is a 
factor of the eliminant of any independent set of four of the 
equations (8) ; second, that in the case of any such set of four the 
solution which corresponds to the discriminant factor of the 
eliminant is indeterminate, and, consequently, that the square of 
the discriminant must be a factor of the eliminant ; third, that the 
discriminant is also a factor of the determinant obtained by 
eliminating x 2 , y 2 , etc., dialytically from the ten equations (8). 
It would certainly be interesting to determine the remaining 
factors of the eliminant of any independent set of four of the 
equations (8), and also the remaining factors of the dialytic 
eliminant of the whole of the equations (8). 
