OF THE ROOTS OP SYSTEMS OP EQUATIONS. 505 
any manner into biparts, which, when all assembled together in a single bracket, are 
represented by 
The symmetric function 0 is expressible as a linear function of assemblages of 
separations of the symmetric function 
38. The theorem of symmetry is as follows :—■ 
Theorem. 
“ When the monomial symmetric function 9^ is expressed as a linear function of the 
assemblages of separations Pg^, . . . Pg^, the coefficient of the assemblage Pg. is the 
same as the coefficient of the assemblage Pg^ when 9^ is so expressed.” 
39. This theorem enables us to form a pair of symmetrical tables in regard to every 
partition of every biweight. The number of tables is therefore twice the number of 
partitions, the generating function for which has been already given. 
§ 6. The Functions comjjosed of One Part. 
40. I will now establish a law by means of which any symmetric function expressed 
by a partition with a single bipart may be at once expressed in terms of separations 
of any partition of its biweight. It is merely necessary to interpret a result already 
obtained. 
I recall the formula of Art. 26, 
(pi)-2 = {¥l) to)n 
which may also be written by Art. 8, 
(_)2.-i (Vtt _ 1)! 
TTj; TTo! . . . 
• • • 
- \PP .r- I .rr 1 
TTi! TTg! 
TTi 7r2 ^ 
Let us compare the cofactor of ... in the development of the left hand side 
with its cofactor on the right hand side. 
When the left hand side is multiplied out each symmetric function product 
TT ^ ^2 
which multiplies the term ... is necessarily a separation of the symmetric 
function {piqf • • •)• The result of the comparison wffil therefore be the 
expression of the function [pq) in terms of such separations. 
41. Let be the value assumed by when and other quantities h are put 
equal to unity. 
Further, let denote the expression of = {pq) by means of separa¬ 
tions of the symmetric function {piqFT> 29 .F • • •)■ 
MDCCCXC.—A. 3 T 
