38 
ME. A. CAYLEY ON AN EXTENSION 
Nothing can be more convenient than the process when the entire series of columns 
is required ; but it is very desirable to have a process for the formation of any column 
apart from the others; and the object of the present memoir is to investigate a rule for 
the purpose. But as Arbogast’s rule, applied as above, depends upon very similar 
principles, I will commence by showing how this rule is to be demonstrated. If we 
take any combination Jfd and operate backwards on the last letter (viz. by changing it 
into its immediate antecessor in the alphabet), we obtain b'^c, which is a term in the 
next preceding column ; ¥d is therefore obtainable from a term in the next preceding 
column, viz. ¥g, and the process is to operate on the last letter. If, instead of ¥d, the 
term is ahd^ (the last letter here entering as a power), the operation backwards on the 
last letter gives a¥G, which is also a term of the preceding column ; and it is to be 
noticed that the last but one letter h is here the immediate antecessor of the last letter c 
(and would have been so even if h had not entered into the given term ab(f, thus axf 
operated on backwards would have given abG). Hence abG"^ is obtained by operating on 
a term in the next preceding column, viz. the term ab'^G, but in this case the operation 
is performed on the last but one letter. Every term is thus obtained from the next 
preceding column, viz. the terms are obtained by operating on the last letter, and (when 
the last but one letter is the immediate antecessor in the alphabet of the last letter) 
then also on the last but one letter, of each term of the next preceding column, and the 
correctness of the rule is thus demonstrated. It is to be observed that the terms are 
operated upon in order, the operation on the last but one letter (when it is operated on) 
being made immediately after that upon the last letter of the same term, and that the 
terms of a column are thus obtained in the proper alphabetical order. 
I pass now to the above mentioned question of the formation of a single column by 
itself ; it will be convenient, by way of illustration, to write down the columns 
aV 
ad^ 
abde^ 
bGe^ 
a(fe^ 
bd?e* 
UGd^e 
&de* 
ad* 
cd?e 
h’^GG* 
d^ 
h^d^e 
b<fde 
bGd? 
c*e 
which belong to the set (a, 5, c, d, e), and are of the degree 6 and the weights 12 and 
15 respectively. 
