486 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
The right hand side of the relation 
(1 + a-^x + (1 -j- a^x -j- ji^y) . . . = I + + • • • + + . . . , 
may be written 
exp («io« + «oi2/)> 
or, since it is convenient to write the symmetric function (pg') in the form this is 
exp (sjoa? + 
where the bar over exp indicates a symbolism by which denotes (10^01^) = «^y. 
Hence the relation 
1 + a^QX + aoip + . . . + apqVfyi + . . . = exp {s^^x + s^^y), 
which is important in connection with the collateral theory of operations to be 
presently brought into view. 
8. Taking logarithms of both sides of the relation 
(1 + a^x + ^^y) (1 + a.^x + /S^y ). . . = 1 + a^gX + cig^y + . . . + a^^x^y^ + . . . , 
there results 
+ W — i + ^02^^) + i (V® + 3%<y + 3s^,xy^ + Sgsy^) 
= log (1 + a^gx + ttgjy + . . . + tfp^xPy^^ + . . .). 
Hence 
1 + a^gX + ctg^y + . . . a^yX^Y + . . . = exp (s^Qa: + Sg^y) 
= exp {5ioa? + .<foiy - h + ^s^xy + SgY) 
+ i (V® + Sszix^y + Ss^^xy^ + Sg^y^) 
Also we have the series of relations :— 
r = a^g, 
^5oi = UqU 
^20 “ ^ 10 ^ ^^20> 
•< 6';^;^ = CtigUgi 
w '^02 ~ ^Ol" 20 . 02 , 
*^91 
’12 
’03 
~ Oio'Of'oi “ ^11^10 “h ^2n 
= <^gi^C<ig ^02^10 ^11*^01 “h ^12’ 
~ OqI^ ^^02^01 d“ ^*^03’ 
* Viz :- 
+ «oi2/ = ^ (“ 1 ^ + di2/) ; + 2 sii*2/ + = 2 (aio; + /3jy)~ ; &c. 
