147] 
OF THE ROOTS OF AN EQUATION. 
419 
Thus, for instance, the combination bd (the partition whereof is 13) contains multiples 
of the two symmetric functions (l 4 ), (21 s ) only. The number of parts in the partition 
13 is 2, and the greatest part is 3. And in the partitions (l 4 ), (21 s ) 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 bd 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 cannot contain a power higher than 2 of any 
root. The reasoning is immediately applied to any other case, and we obtain 
First Restriction.—A combination b v c q ... contains only those symmetric functions 
(l x m y ...), for which the greatest part does not exceed the number of parts in the 
partition 1^2 5 ... , and the number of parts is not less than the greatest part in the 
same partition. 
Consider a partition such as 1 2 2, then replacing each number by a line of units 
thus, 
1 
1 
11, 
and summing the columns, we obtain a new partition 31, which may be called the 
conjugate 1 of 1' 2 2. It is easy to see that the expression for the combination b 2 c (for 
which the partition is 1 2 2) contains with the coefficient unity, the symmetric function 
(31). the partition whereof is the conjugate of 1 2 2. In fact b~c = (— Sa) 2 (£a/3), which 
obviously contains the term + la 3 /?, and therefore the symmetric function with its 
coefficient +1(31); and the reasoning is general, or 
Theorem. A combination b q c q ... contains the symmetric function (partition conjugate 
to 1^2®...) with the coefficient unity, and sign + or — according as the weight is even 
or odd. 
Imagine the partitions arranged as in the preceding column, viz. first the partition 
into one part, then the partitions into two parts, then the partitions into three parts, 
and so on; the partitions into the same number of parts being arranged according to 
the magnitude of the greatest part (the greatest magnitude first), and in case of 
equality according to the magnitudes of the next greatest part, and so on (for other 
examples, see the outside column of any one of the Tables). The order being thus 
completely defined, we may speak of a partition as being prior or posterior to another. 
We are now able to state a second restriction as follows. 
Second Restriction.—The combination №c q ... contains only those symmetric functions 
which are of the form (partition not prior to the conjugate of iv2 q ...). 
The terms excluded by the two restrictions are many of them the same, and it 
might at first sight appear as if the two restrictions were identical; but this is not 
1 The notion of Conjugate Partitions is, I believe, due to Professor Sylvester or Mr Ferrers. [It was due to 
Mr now Dr Ferrers.]