512 
MAJOR MACMAHOR ON SYMMETRIC FUNCTIONS 
We have, in fact, a biweight operator decomposed into a full number of bipart 
partition operators 
g oi^^oi _ _ _ .. .)• 
Observe that the whole theory of separations is a generalisation from a weight to a 
partition of a weight. Here we have generalised from a weight operation to a 
partition operation, and I henceforward regard the partition operator as the essential 
linear partial differential operator of the theory. The biweight operator gj,^ has been 
expressed ante (Art. 17) in term.s of the obliterating operators of the form Gp^. These 
operations are equally available in the theory of the separable partition in general. 
The mode of their operation upon a symmetric function product will be subsequently 
explained (in §11). So far I have merely considered their operation upon monomial 
forms.* 
53. I observe that the biweight operator gp^^ is expressed as a linear function of the 
partition operators of the same biweight, according to the same laws as— 
(1) The operator gp^ is expressed in terms of the operations Gpq (Art. 17). 
(2) The symmetric functions, containing one part only, are expressed in terms of 
the fundamental or single-unitary forms (Art. 8). 
E.g., compare the three results (the first slightly modified) :— 
/ \ j. + o GStt — 1)! 
-pi q\ 
TT. ! TTo ; . . 
p ! ? 1 "" TTi! TTa !.. 
p! (2' 
TT, ; TTo! . . . 
^Pi'h 
54. For convenience of reference, I write down the particular simplest cases of the 
decomposition. 
t/io — g{To)i 
901 = 
9-20 — 9{m ^9c^)’ 
9ii — t/(TooT) ^7(n)j 
902 = t 7 ( 0 T 2 ) 2 ^( 05 )) 
t/so — 17(103) “ 317(2 o To) “b ‘^9(So)y 
921 — 9(iQ- 01; 17(20 To; ,9'(TT TO) "b 1/(2T)> 
17i 2 — l/( 0 l 2 10 ) 1/(02 To) ^(TT OT) T 9 ( 12 )} 
903 — ^(oi3) 3^(55 -fi Sg(^y 
* The decomposition of the obliterating operation into partition obliterating operations is given 
post § 10. 
