1898-99.] Mr E. J. Nanson on a Set of Quadrics. 
353 
On the Eliminant of a Set of Quadrics, Ternary or Qua- 
ternary. By E. J. Nanson. Communicated by Professor 
Chrystal. 
1. There are two methods for expressing in determinant form 
the eliminant of three ternary quadrics. If u , v, w are the quadrics 
and J their Jacobian we may eliminate dialytically the ten 
quantities x 3 , y 3 , z 3 , y 2 z, yz 2 , z 2 x, zx 2 , x 2 y, xy 2 , xyz from the ten 
cubics xu , yu, zu, xv, yv , zv, xw, yiv , ziv, J. This is in effect the 
process given by Sylvester in 1841.* The other method is to 
eliminate dialytically the six quantities x 2 , y 2 , z 2 , yz, zx, xy from 
u, v, w and the three differentials of the Jacobian. t 
2. The eliminant of four quaternary quadrics may be found in 
determinant form by a process having points in common with each 
of these two methods. Multiplying each of the quadrics by the 
variables x, y, z, iv in turn we get sixteen quaternary cubics, 
and we have also the four differentials of the Jacobian. Thus we 
have in all twenty quaternary cubics, or just sufficient to eliminate 
dialytically the twenty expressions x 3 , ... , x 2 y, . . . xyz, . . . 
The result is a determinant of order 20, and of the correct degree, 
viz., 8, in the coefficient of each quadric. 
It does not appear to be possible to extend this process so as to 
obtain, in determinant form, free from extraneous factors, the 
eliminant of a set of quadrics in more than four variables. 
3. As an illustration of the two methods of forming the elimi- 
nant of a set of ternary quadrics consider the equations 
discussed by Sylvester J and Muir. § 
* Cambridge Mathematical Journal , vol. ii. p. 235. See also Muir, 
History of Determinants , p. 233. 
+ Salmon, Higher Algebra , § 90, p. 85. 
$ Loc. cit., p. 233. 
§ Proc. Roy. Soc. Edin. , vol. xxi. pp. 220-234. 
(Read January 9, 1899.) 
ax 2 + fyz -f- gzx+ iixy = 0 
by 2 + fyz + g'zx + li'xy — 0 
cz 2 + f"yz + g'zx + h"xy = 0 
