490 
MR. A. CAYLEY ON THE SY^DIETRIC ET:NCTI0N.S 
e h(l & Ifc If (line) 
4 13 2^ 1^2 
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 ^) 
(2r) 
( 1 ^), _ 
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 fu'st 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 
(F), 
1 
4 
2 
3 
id 13 
(F), (21T, 
2 
3 
2 
2 
2' 
(lA (21-’), (2T, 
2 
2 
3 
2 
1-2 
(Iri, (21ri, (2=), (31), 
3 
2 
4 
1 
(V), (2f), (2-’), (31), (4) 
1 
Thus, for instance, the combination hd (the partition whereof is 13) contains multiples 
of the two symmetric functions (1^), (21^) only. The number of parts in the partition 
13 is 2, and the greatest part is 3. And in the partitions (I”*), (21') the greatest part 
is 2, and the number of parts is not less than 3. The reason is obvious ; each term of 
the developed expression of hd must contain at least as many roots as are contained in 
each term of d, that is 3 roots, and since the coefficients are linear functions in respect 
to each root, the combination bd camiot contain a power higher than 2 of any root. 
The reasoning is immediately applied to any other case, and w'e obtain 
First Restriction. — A combination b^c'^... contains only those symmetric fimctions 
for which the greatest part does not exceed the number of parts in the parti- 
tion R2’.., and the number of parts is not less than the greatest part in the same 
partition. 
Consider a partition such as 1^2, then replacing each number by a line of units thus, 
1 
1 
11 , 
