504 
MAJOR MACMAHON ON SYMAJETRIC FUNCTIONS 
in the abbreviated notation 
Cfl = . . . -j- P(,6,. + • • • 
We have then the relations :— 
Ce, — • • ■ + • • • (/b = 1, 2, 3, . . ./l'), 
and 
4 " 4“ • • • “b 'n'l^k^ki 
P^t = + • • • + 
the quantities m being numerical. 
The determinant of the numbers m is symmetrical, for by the law of symmetry 
m.,, = m,,. 
Hence, the coefficient of symmetric function 9^ in the development of the assemblage 
of separations P^,, is identical with the coefficient of symmetric function 6,. in the 
development of the assemblage of separations Pg,. 
We may now proceed to express the symmetric functions ^as linear functions of the 
assemblages of separations Pg and by elementary theory of determinants, the deter¬ 
minant of the system of results is symmetrical. Hence 
^1 — "b 1^12^0, + • • • + 
^■2 — + • • • + p-’/Hflo 
= H-kiPe^ + P'X.aP^a + • • • 4“ 
wherein 
f^rs - P'S/ * 
36. From this are deduced two im})ortant theorems, the one a theorem of expressi- 
bility, and the other a theorem of symmetry. Any one of the monomial symmetric 
functions 9 is expressed by a partition which is a specification of a separation of the 
partition • • •)• This implies that the biparts occurring in the partition of 9 
can be so partitioned into biparts that when assembled together they will be 
identical with the biparts of the partition • • •)• Hence the theorem of 
expresslbility :— 
37. TJi eorem. 
“ The biparts of the partition of a monomial symmetric function 9 are partitioned in 
