506 
MAJOR MAOMAHON ON SYMMETRIC FUNCTIONS 
Then we may write 
TT, ! TTc! . . . ViTLi'^ ' ' ') 
(Stt — 1)! 
TTilTTa!... • ■ • 
where denotes the sum of all the symmetric functions of hi weight pq. 
Represent the different separates of the partition { piqi^P-iq-i '^- • •) by (J^), (J 3 ), . . . 
and any separation by • • 5 substitute for the quantities their values 
in terms of symmetric functions; apply the multinominal theorem and equate corre¬ 
sponding portions of the two sides and there results the formula 
(Xtt - 1) ! 
TT^! TTj! . . . • • •) 
(sy-D! 
Jl-J2- • • • 
. . . 
where the summation is taken for every separation of the given partition. 
42. This important result is a gener-alisation of the Vandermonde-Waring law 
for the expression of the sums of the powers of the roots of an equation in terms of 
the coefficients. 
43. The formula may be reversed so as to exhibit any symmetric function whatever 
in terms of single bipart functions. The result easily reached is 
(7wr‘ ■ ■ ■) 
— 1)! {S'7r2 — 1)! . . . 
. . . TT^! TTjg! . . . TTo]^! TTjo! , . . i>i22iY'“ • ■ • ) . .) ' ’’ 
the summation being for every separation 
of the symmetric function 
§ 7 . Second Law of Symmetry. 
44. The operation 
Q — - 1 - fOn 0„ - 1 - Ctnl 0(, -f- . . . -f- a,-x 0« “k • • • 
may be said to be of biweight pq, since it lowers the weight of a symmetric function 
by the biweight Further, its degree is zero, since it does not in general lower 
the degree of a symmetric function. If, however, gpq operates upon a-symmetric 
function of its own biweight, it is equivalent to the simple differential operation da„, 
and is of degree unity. 
