510 
MAJOR MACMAHON ON SYAIMETRIC FUNCTIONS 
§ 9. The linear Partial Differential Operations of the Theory of Separations. 
49. For purposes of calculation it is necessary to adapt the operations 
fJoa • • • 9v<i ■ • •’ 
so that they may be performed on a symmetric function when the latter is expressed 
in terms of the separations of any given partition. 
Of any partition {piyPp^eip . . .) separates {vide Definitions) are formed by taking 
all possible combinations of the parts. These are precisely 
(tti -h 1) {'^z + 1) . . . —1, 
distinct separates which must be regarded as independent variables. 
Put 
^I0’^io+Pio01’"oi+Poi ^ . . .) 
for any separate of a given separable partition P. 
Then by a known theorem 
>i5i+^Pi9i . . .)) 
the summation being in regard to all the separates. 
Moreover (Art, 17) 
!“■/ ..I ..I 9pri — ^ , —::—r t^ioI joi • • • 
p! gl 
'^10 i '^01! • • • TT"' 
PiQi ■ 
the summation being in regard to all the partitions (1P"'. . . • • •) ‘^^ 
biweight ; and also (Art. 13) 
-10 -01 ^PlQl _ _ __ ___ __ 
GiyGoi. . . . . . (10’'‘'''^'’“’01"'"''^'’'” . . (10'’'°01'’“ . . .pi<]P^^^^. . •) • 
50. Hence 
' / oo ! I 
p\ q\ . . 
9p 
(—ASii-l /Xtj- _ IV__ 
= ^—p-p— (KF^oi' 
“-^p .^11 
^10- ’^01- 
' Pl<ll • • • • 
• T dll • • •) ^(lo-'io+Poioi Oi + Pol _ _ _ ^iqjirPiSi + PiijSi , . .) 
the summation being in regard to 
(1) Every separate of the given separable partition ; 
