494 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
is in co-relation with the symmetric function 
ilhai' • • • ) • 
19. There is thus complete correspondence between quantity and operation, and 
any formula of quantity may be at once translated into a formula of operation. 
Observe that a product of symmetric functions 
• • • ) O’rSi'” • •) 
is in corresjDondence with the operation 
1 1 
TT"! ! TTo ! 
1 1 
Pu pi^- 
9r^r9r^:^ 
the notation indicating that the two operations 
1 1^ 
Pi! Pi'-' 
9r^:^9r^t 
and 
ri ! TTj! 
9ppi:'9p^i: 
are to be successively performed. 
For an example take the algebraic formula 
(31 01) = - 1 (21 10) (01) + 1 (21 01) (10) + i (10 01) (21) - l (21 10 01), 
which is translated into the operator formula 
93i9oi = - i 9-2i9io • 17 oi + h 92i9oi • S'lo + i 9w9oi • 9^1 ~ h 92i9io9oi- 
20 . It is now necessary to enquire into the laws which appertain to the performance 
of these operations upon symmetric functions. We have seen, ante Art. 13, the law 
by which the obliterator is performed upon a monomial symmetric function. 
Since 
(-) 
p + rj 
-1 (p + g - 1)! _ 
2P q\ 
9p'j 
..-GSvr-l)! 
Tr.2 
we can operate with gp,j upon a monomial form by operating independently with the 
successive G products on the right, and adding the results together. As a particular 
result, observe that a term on the right is G;;^, and hence 
^ 9p^a-j = ^P'M = in) = ]’ 
or 
9p'^pq — ( 
I rt' 
p\q 
(qj + (j — 1)' 
