420 
A MEMOIR ON THE SYMMETRIC FUNCTIONS 
[147 
so: for instance, for the combination bd 2 , see Table YII (a), the term (41 3 ) is excluded 
by the first restriction, but not by the second; and on the other hand, the term 
(3 2 1), which is not excluded by the first restriction, is excluded by the second restriction, 
as containing a partition 3 2 1 prior in order to 32 2 , which is the partition conjugate 
to 13 2 , the partition of bd 2 . It is easy to see why bd 2 does not contain the symmetric 
function (3 2 1); in fact, a term of (3 2 1) is (a 3 /3 3 Y), which is obviously not a term of 
bd 3 = (— 2a) (2a/3y) 2 ; 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 Arbogast’s Method of Derivations 
from an operand a x , a being ultimately replaced by unity. The squares above the 
sinister diagonal are empty (i.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 greater part of the numbers in the Tables (a) were calculated, those of each 
table from the numbers in the next preceding table by the following method, 
depending on the derivation of the expression for b v+1 c q ... from the expression for b p c q ... 
Suppose, for example, the column cd of Table Y (a) is known, and we wish to calculate 
the column bed of Table YI (a). The process is as follows: 
Given 
we obtain 
2 2 1 
21 3 l 5 
1 
3 10 
321 
2 3 
31 3 
2 2 1 2 
21 4 
1« 
1 
3 
3 
2 
6 
12 
10 
60 
1 
3 
3 
8 
22 
60 
where the numbers in the last line are the numbers in the column bed of Table 
YI (a). The partition 2 2 1, considered as containing a part zero, gives, when the parts 
are successively increased by 1, the partitions 321, 2 3 , 2 2 1 2 , in which the indices of the 
increased part (i.e. the original part plus unity) are 1, 3, 2; these numbers are taken 
as multipliers of the coefficient 1 of the partition 2 2 1, and we thus have the new 
coefficients 1, 3, 2 of the partitions 321, 2 3 , 2 2 1 2 . In like manner the coefficient 3 of