534 MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
quantities h and symmetric functions of the quantities a, (3. These are given in 
Art. 24. 
To put the group in evidence it is necessary to modify these relations by writing 
Pi 
for so that for examples the expressions for and become 
(20) ha + (To’) he, 
(TT) hi,+ (rooT)h,Ai 
respectively. In any product cofactor of the product 
1 ''i ''a 
is composed of symmetric function products each of which appertains to the group 
Gn {{rpp^'p {rppf^ . . ., . . 
■) 
i- 
The sum of the coefficients attached to the members of this group is obtained by 
putting each monomial symmetric function equal to unity. The sum in question then 
ajdpears as the numerical coefficient of the h product above written. 
Write then 
cw = K 
— ^01 
— ^-^30 “h ^10=’ 
-h ^10^01 
so that, ^ and g being arbitrary, 
i + Cjy‘ ^ 17 -j- . . . + + ■ • • 
— (1 + ^10^ + L q- -f + ...)... 
a factor appearing on the right for each biweight. 
To find the sum of tiie coefficients in each groujr in the case of the expression of the 
single-bipart functions we have now merely to take logarithms when (Art. 26) 
the functions being 67,^ or {pq), the sum in each case presents itself multiplied by 
(— ( 1 /a ^ 7 0 (a + 7 ~ 1) •• Expanding the right hand side after taking logarithms 
we see that only terms of the form 
(Ti (T -2 
Zr.jPi hi-gPi . . 
can appear. Hence the theorem :■ 
