OF THE EOOTS OP AN EQUATION. 
491 
and summing the columns, we obtain a new partition 31, which may be called the con- 
jugate* of 1^2. It is easy to see that the expression for the combination Ifc (for which 
the partition is 1^2) contains with the coefficient unity, the symmetric function (31), the 
partition whereof is the conjugate of 1^2. In fact i^c=(— 2a)^(2a/3), which obviously 
contains the term +la®j3, and therefore the symmetric function with its coefficient 
+ 1(31); and the reasonmg is general, or 
Theorem. A combination .. 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 h^(fl . . . contains only those symmetric functions 
which are of the form (partition not prior to the conjugate of +2^...). 
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 so : for 
instance, for the combination see Table VII(«), the term (41^) is excluded by the 
first restriction, but not by the second; and on the other hand, the term (3^1), which is 
not excluded by the fii’st restriction, is excluded by the second restriction, as containing 
a partition 3^1 prior in order to 32^, which is the partition conjugate to 13^, the partition 
of h(B. It is easy to see why hd? does not contain the symmetric function (3^1); in fact, 
a term of (3^1) is (a^(3+), which is obviously not a term of hd^ — \ but I 
have not investigated the general proof. 
I proceed to explain the construction of the Tables (a). The outside column 
contains the symmetric functions arranged in the order before explained ; the outside or 
top line contains the combinations of the same weight arranged as follows, viz. the 
partitions taken in order from right to left are respectively conjugate to the partitions 
in the outside column, taken in order from top to bottom ; in other words, each square 
of the sinister diagonal corresponds to two partitions which are conjugate to each other. 
It is to be noticed that the combinations taken in order, from left to right, are not in 
the order in which they would be obtained by Aebogast’s Method of Derivations from 
an operand a being ultimately replaced by unity. The squares, above the sinister 
diagonal are empty (^. e. the coefficients are zero), the greater part of them in virtue of 
both restrictions, and the remainder in virtue of the second restriction; the empty 
squares below the sinister diagonal are empty in virtue of the second restriction ; but 
the property was not assumed in the calculation. 
* The notion of Conjugate Partitions is, I believe, due to Professor Stltestee or Mr. Feerebs. 
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