418 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 
where as usual the summation extends only to the distinct terms, so that e.g. (p 2 ) 
contains only half as many terms as (pq), and so in all similar cases, then we have 
-b = (l), + c = (l 2 ), — d = (l 3 ), &c. ; 
and the two problems which arise are, first to express any combination №&... in terms 
of the symmetric functions (l x m y ...), and secondly, or conversely, to express any 
symmetric function (l x m y ...) in terms of the combinations №c q ... . 
It will conduce materially to brevity if 1*22... be termed the partition belonging 
to the combination b p c q ... ; and in like manner if l x ni y ... be termed the partition 
belonging to the symmetric function (l x m y ...), and if the sum of the component 
numbers of the partition is termed the weight. 
Consider now a line of combinations corresponding to a given weight, e.g. the 
weight 4, this will be 
e bd c 2 b-c i 4 (line) 
4 13 2 2 1 2 2 T, 
where I have written under each combination the partition which belongs to it, and 
in like manner a column of symmetric functions of the same weight, viz. 
(4) (column) 
(31) 
(2 2 ) 
(2P) 
(l 4 ), 
where, as the partitions are obtained by simply omitting the ( ), I have not separately 
written down the partitions. 
It is at once obvious that the different combinations of the line will be made up 
of numerical multiples of the symmetric functions of the column ; and conversely, that 
the symmetric functions of the column will be made up of numerical multiples of the 
combinations of the line ; but this requires a further examination. There are certain 
restrictions as to the symmetric functions which enter into the expression of the com 
bination, and conversely, as to the combinations which enter into the expression of the 
symmetric function. The nature of the first restriction is most clearly seen by the 
following Table: 
Number of 
Parts. 
Greatest 
Part. 
Combinations 
with their several 
Partitions. 
Contain Multiples of the 
Symmetric Functions. 
Greatest Part 
does not exceed 
Number of 
Parts not 
less than 
1 
4 
e 4 
(I 4 ), 
1 
4 
2 
3 
bd 13 
(l 4 ), (21 2 ), 
2 
3 
2 
2 
c 2 2 2 
(l 4 ), (21 2 ), (2 2 ), 
2 
2 
3 
2 
b 2 c 1 2 2 
(lfi, (21 2 ), (2 2 ), (31), 
3 
2 
4 
1 
Z> 4 l 4 
(l 4 ), (21 2 ), (2 2 ), (31), (4) 
4 
1