488 
MAJOR MACMAHON OR SYMMETRIC PURCTIORS 
an identity which arises from the former by interchanging the letter h with the 
letter a. 
Hence, if /‘and be any two functions, such that 
then also 
0 
0 
11 
0 
0 
• • l^pqi ' • 
•1 
(/) (ct^oj • • 
11 
0 
0 
• • i^pqi • • 
and, in general, in any relation connecting the functions a with the functions h, an 
identity will still remain if the letters a and li be transposed. 
By the multinomial theorem 
M-f 
,77 — 1 
TtP TTg! . . . 
TTj TTg 
• • • 
From a previous result in this article, by taking logarithms and expanding 
(p + g - 1 )! 
p\ q\ 
^pq 
i'l'K — IF 
— V _hh 7, 7, 
V / TT I TT I 
" 1 • " 2 • • • • 
which is to be compared with the formula 
(-) 
p\ g! 
= 2 ,(-p 
-1 
(Stt - 1 )! 
TT]^! TTg! . . . 
’Tl 
^Pi<li 
and it will be noticed that Sp^ remains unchanged when li is written for a, except for 
a change of sign, when the weight + 5 ' is even. 
§ 3. TJie Differential Operations. 
1''2. The beautiful properties of these symmetric functions are most easily established 
by means of the differential operations whose theory I proceed to establish. 
Consider the identity 
(1 + a^x + /3jy) (1 + a.^x + ^.pj) . . . (l -f a,,a: + ^„y) 
= 1 + + cCqPj + a.Qx" + a^yvy + a^ff 
where n may be as large as we please. 
Multiply each side by (l + p,ft: + py). 
The right hand side becomes 
1 + (^ho + /^) a; + («oi + r) y + -f ya^^) x~ + + ya^^ + va^^) xy 
+ (^02 + ^«'oi) 2 /^ + • • • 5 
