101 
733] 
then, if 
we have 
ON A FORMULA OF ELIMINATION. 
©i = 0, b, C, d\6 u l) 3 , 
© 2 = (a, b, c, d$0 a , l) 3 , 
V = .d 3 ©^ ; 
viz. on the right-hand side, replacing the symmetrical functions of 6 1 , 0. 2 by their 
values in terms of A, B, G, we have the expression of V in its known form 
V = a 2 G s 4- &c. 
Form now the expressions 
©i-©2, 02®1-^1©2, ^©! - ^©2, 
each divided by 0 1 — 0 2 . These are evidently symmetrical functions of 0 1} 0 2 , the 
values being given by the successive lines of the expression 
0, 
1, 
0j + #2, 
0 2 + 0j0 2 -\- 0 2 
$e£, 3c, 36, a) ; 
-1, 
0, 
OA, 
0\0z (0i + 0-i) 
- ($1 + $2), 
- OA* 
0, 
W 
— {@1 + ^1^2 + $2 2 )> 
— 0 Y 0 2 (0 1 + # 2 ), 
- №, 
0 
and, consequently, these same quantities, each multiplied by A 2 , are given by the 
successive lines of 
0, 
A 2 , 
-2AB, 
— AG + 4<B^d, 3c, 3b, a). 
-A 2 , 
0, 
AC, 
-2BG 
2 AB, 
— AG, 
0, 
G 2 
AG - 4B 2 , 
2BG, 
-c\ 
0 
Calling these X, Y, Z, W, that is, writing 
X = 3A 2 c — 6ABb + (— AC+ 4B 2 )a, &c., 
then X, Y, Z, W are the values of 
© x -© 2 , dj&l- 0!©., 6»2 3 ©1-^1 3 © 2 , 
each multiplied by A 2 -s- (0\ — 6%) ; and the functions all foui of them vanish if only 
@ 1 = 0, ©2 = 0; or, what is the same thing, the equations X = 0, F=0, Z=0, W=0 
constitute only a twofold system. 
The functions 
( X, Y, Z )