OF THE ROOTS OP SYSTEMS OF EQUATION'S. 
535 
“ In the expression of symmetric function [pq) by means of separations of any 
partition of the same biweight, the partition consisting of dissimilar parts, the 
algebraic sum of the coefficients in each group of separations is zero.” 
As regards the remaining cases where the separable ^^artition does not contain 
dissimilar parts, the group obviously contains but a single separation and qua group 
has no existence. 
We have in fact the expression of [pq] by means of separations of {rs'‘) where 
Ir = p>, ^’5 = q- 
The result is clearly 
(-) 
'p + q-l 
-1 + !? — 1 )' /— 
pi cp. 
{pq) = ^i-r 
Scr - 1 1 
^,/T - 
1)! 
79. The law of group of separations may be verified from the tables. It is a very 
satisfactory aid to calculation, particularly in the detection of missing separations. 
Moreover the law embraces symmetric functions other than those symbolised by a 
single bipart. Suppose the function expressed in terms of single-bipart functions. 
The latter may be separately expressed in terms of separations of partitions in such 
wise that the function in cpuestion will be represented by means of separations 
of any given partition of its biwelght. The law of the group will hold for the 
single-bipart functions whenever the separable partition contains dissimilar parts, and 
moreover, in a product of single-bipart functions the law will hold if one or more of 
the factors is expressed in terms of separations of a partition containing dissimilar 
parts. Hence the only exception occurs when we find presented a product of the 
form 
• • • 5 
now if the symmetric function, say {pPlPpgqp • • •)> whose expression we a,re 
considering in connection with a given separable partition, say {ci-J)p ((-P.p . . .), itself 
possesses a separation of specification 
{apt,, • ■ •) 
a product of this form will certainly occur, but not otherwise. 
Hence the theorem :— 
80. “ In the expression of symmetric function 
by means of separations of 
the algebraic sum of the coefficients in each group of separations is zero if the partition 
