OP THE ROOTS OF SYSTEMS OP EQUATIONS. 
501 
and performing each side of this equation upon the opposite side of the equation (A) 
we obtainj after cancelling of 
no other terms surviving the operations, or 
L J ; 
since the symmetric function on either side is reduced to unity by the operation. 
Similarly the equation (C) yields the equation of operators 
Pi P2 
+ M {piq{^Pi,qp . . .) G",.^.,^. . . + 
and this when performed on opposite sides of equation (A) gives 
Hence 
M = J. 
L = M, 
and we have the law of symmetry expressed by the two relations 
_ _ Pi P2 
. . . + L {'piqP'JVlz' • • •) 
viz., if in the first of these relations the partitions {piqP^p>zqz '' • • •)’ • • •) 
be interchanged the numerical coefficient L remains unaltered. 
The theorem is a consequence of the two properties that have been established in 
regard to the three identities and the relation assumed to exist between the quantities 
involved in them. 
It appears to be the most important theorem in symmetrical algebra. 
33. I now pass to certain consequences v/hich flow straight from the theorem. 
It is necessary to make a few definitions. 
Definition. 
“ A partition is separated into separates by writing down a set of partitions, each 
separate partition in its own brackets, so that when all the parts of these partitions 
are assembled in a single bracket, the partition which is separated is reproduced.” 
Giii ^Piii • • • 
C„j,^ ... — 
