492 
MAJOR MACMAHON ON SYMMETRIC EUNCTIONS 
16. To apply it to tlie case in hand, put 
«iio = p, tiIqi — V, mjjrj = 0 in other cases. 
Then 
+ M-qi't) + . . . + + • • • = log (1 + 
and the result is 
exp (p^'io + vffoi) = exp + vg^^ - i (Yi/.o + 
-f 3 {l^^9zo + + 3/xz/2^i3 + —...}. 
Combining with a former result 
l^9io + ^5^01 — 2 (/^Vso + + v~g^^2) + ¥ (/^Vso + 1 /^ 03 ) — • • • 
= log (1 + /xGlo + + . . . + /A^VGy,y +...). 
(Compare Art. 8.) 
17 . By expanding the right hand side we can express each linear operator of the 
form gj,fj in terms of products of the obliterating operators which have the form Gj,^. 
The law^ is identical wdth that which expresses the functions containing one bipart 
in terms of the fundamental symmetric functions. We find 
9w — ^10 
9oi — ^ui 
r 9-20 = “ 2Goo 
9n = GlojGoi — Gj^ 
L1/o2 ~ 2 G 02 
fl/so = Gli,/ — SClouGrio + 3G00 
9-2] — ^ 10 “ ^ui — ClooGoi — GioGio + Goi 
, 9i2 = ~ Gro^Gjo — Gi,G,|i + G^3 
L< 7 o 3 = <^ 01 ^ — 3G03G01 + 3 Go 3 
and in efeneral 
while 
\ ) .p\q\ 9pq—^-\ ) ^ ■■■ 
' 1 ■ " 2 • • 
^ ^ ' 2h'- 9i'- J 1 Ih' 
(-) 
TT, ! TTo ! . 
— 1 TTi TTj 
9piqi 9piV-2 
(Compare Art. 8.) 
