1897 - 93 .] Mr E. J. Hanson on a Problem of Sylvester s. 
153 
the fact that the dialytic eliminant has A 2 as a factor by con- 
sidering (1) as a special case of the more general system 
bz 2 + e'y 2 - 2 xyz = 0 
l . 
( 7 ) 
ex 2 + a'z 2 — 2gzx = 0 
ay 2 + b'x 2 - 2hxy = 0 J 
But in point of fact his work shows that the true eliminant of 
(1) is not A 2 hut a 2 5 2 c 2 A 2 . It is not, however, necessary to 
consider a system more general than (1) in order to arrive at this 
result. 
7. In the first place, if, as explained in Salmon’s Higher 
Algebra , we eliminate dialytically from (1), and from the three 
equations obtained by equating to zero the differential coefficients 
of the factors of A, B, C, we get the eliminant in the form 
. 
c 
b 
-2/ 
c 
. 
a 
. 
-2 g 
b 
a 
. 
-2 h 
cF 
bF 
2 (abc -fgh) 
2bR' 
2cG' 
cG' 
aG' 
2aR' 
2 {abc -fgh) 
2cF' 
6H' 
dQ! 
. 
2aG' 
2 bF 
2 (abc -fgh) 
and reducing the determinant exactly as in § 2 we get 
c b 
8abc . c . a 
b a 
that is 16 aWA 2 . 
8. In the second place, it may be shown by elementary algebra 
A' 
H' 
G' 
Eh 
B' 
F 
G' 
F 
C' 
that the eliminant is a 2 b 2 c 2 A 2 . For we have in 
nate x, y, z from 
effect to elimi- 
ex 2 + az 2 - 2 gzx = 0 
• • («) 
ay 2 + bx 2 - 2 hxy = 0 
• ■ W) 
my + nz = 0 
(y) 
where my + nz is a factor of bz 2 + cy 2 - 2 xyz, 
so that 
bm 2 + en 2 + 2 xmn — 0 
• • (8) 
How eliminating z from (a), (y), we have 
cn 2 x 2 + nm 2 y + 2 gmnxy = 0, 
YOL. XXII. 
14 / 9/98 
