146 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
and the fresh departure consists in viewing such an equation as the out- 
come of the set of equations 
x + y + z - f . . . = 0 , x 1 — a , y 1 = b , z 2 = c , .... 
Taking the case where the number of variables in the set is three, Cayley 
operates on the equation x+y+z= 0 with the multipliers 1 , yz , zx , xy , 
thus obtaining with the help of the other equations a set from which the 
variables x , y , z , xyz may be eliminated with the result * 
.111 
1.6b 
1 6 . a 
1 b a . 
Also, and, so to say, conversely, he operates with the multipliers x ,y ,z , xyz , 
and eliminates 1 , yz , zx , xy , with the result 
. a b c 
a . 1 1 
b 1.1 
g 11 . 
= 0. 
Similarly, when there are four variables, the multipliers are 
1 , yz , zx , xy , xio , yw , zw , xyzw , 
and the eliminands 
x, y , z , w , yzw , zwx , wxy , xyz , 
or vice versa ; but in this case the two resultants are not essentially 
distinct, the one being derivable from the other by mere transference of 
lines. Cayley then adds, “ And in general for any even number of 
quadratic radicals the two forms are not essentially distinct, f but may 
be derived from each other by interchanging lines and columns, while for 
an odd number of quadratic radicals the two forms cannot be so derived 
from each other, but are essentially distinct.” 
The equations next dealt with are of the type 
a 1 + b i + c i + . . . . = 0 , 
Sylvester having suggested the extension of the process. Taking the case 
of three variables, that is to say, when the set of equations is 
x + y + z = 0 , x^ = a, y 3 = b, z B = c, , 
* Already thus formed by Cayley in his first paper of all (1841). 
t Observe that, although Cayley considers the two determinants of the previous case to 
be essentially distinct, the second is derivable from the first by multiplying the columns by 
abc , a, b , c respectively, and then dividing the rows by 1 , be , ca , ab respectively. 
