492 
ME. A. CAYLEY OX THE SY^klMETEIC EEXCTIOXS 
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 from the expression for ... Suppose, 
for example, the column cd of Table V(a) is knovm, and we wish to calculate the 
column bed of Table VI(a). The process is as follows : — 
Given 
we obtain 
1 
1 
V 
1 1 
3 
10 
321 
23 
31® 
2=1- 
2H 
1 
3 
2 
3 
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 Vl(ff). 
The partition 2^1, considered as containing a part zero, gives, when the parts are success- 
ively increased by 1, the partitions 321, 2®, 2^1^, in which the indices of the increased 
part {i. e. the original part plus unity) are 1, 3, 2 ; these numbers are taken as multi- 
pliers of the coefficient 1 of the partition 2^1, and we thus have the new coefficients 
1, 3, 2 of the partitions 321, 2®, 2^1^. In hke manner the coefficient 3 of the pai-tition 
21^ gives the new coefficients 3, 6, 12 of the partitions 31®, 2®1®, 21^, and the coefficient 
10 of the partition 1® gives the new coefficients 10, 60 of the partitions 21^ and 1®. and 
finally, the last line is obtained by addition. The process in fact amounts to the multi- 
plication separately of each term of ed~ 
l(2®l)-|-3(2r)-l-10(l®) 
by 6=(1). It would perhaps have been proper to employ an analogous rule for the 
calculation of the combinations e'^d’' .. not containing b, but instead of doing so I availed 
myself of the existing Tables (J). But the comparison of the last line of each Table {a) 
(which as corresponding to a combination b^ was always calculated independently of the 
Tables {b)) with such last line as calculated from the corresponding Table [b), seems to 
afford a complete verification of both the Tables ; and my process has in fact enabled me 
to detect several numerical errors in the Tables (i), as given in the Enghsh translation 
of the work above referred to. It is not desirable, as regards facility of calculation and 
independently of the want of verification, to calculate either set of Tables wholly from 
the other ; the rules for the independent calculation of the Tables {b) are fully and clearly 
explained in the work referred to, and I have nothing to add upon this subject. 
The relation of symmetry, alluded to in the introductory paragraph of the present 
memoir, exists in each Table of either set, and is as follows : \lz. the number in the Table 
corresponding to any two partitions in the outside column and the outside line respect- 
ively, is equal to the number corresponding to the same two partitions in the outside 
line and the outside column respectively. Or, calling the two partitions P, Q, and 
\\uiting for shortness, combination (P) for the combination represented by the partition 
