[ 37 ] 
II. On an Extension (^Arbogast’s Method of Derivations . 
By Arthur Cayley, Esy.^ F.B.S. 
Eeceived October 18, — Head December 13, 1860. 
Arbogast's Method of Derivations was devised by him with a view to the development 
of a function <p(a-\-bx-\-cx ‘-\- but it is at least as useful for the formation of only 
the literal parts of the coefficients, or, what is the same thing, the combinations of a 
given degree and weight in the letters (a, b, c, d, . . .), the weights of the successive letters 
being 0, 1, 2, 3, &c. Thus instead of applying the method to finding the coefficients 
a*, 4:03, &c., 
we may apply it merely to finding the sets of terms 
a*, a^b, cdc, &c. 
a^V^ 
To derive any column from the one which immediately precedes it, we operate on a 
letter by changing it into its immediate successor in the alphabet, and we must in each 
term operate on the last letter, and also, when the last but one letter in the term is the 
immediate antecessor in the alphabet of the last letter (but in this case only), operate 
on the last but one letter. Thus c^c gives (Fd, but (Fb"^ gives cC-bc and alf, and the next 
succeeding column is therefore 
o?d 
a^bc 
aE. 
If the series of letters is finite, and the last letter of the term is also the last letter of 
the series, then it is impossible to operate on the last letter of the term, but the last 
but one letter (when the foregoing rule applies to it) is still to be operated on ; and if 
the rule does not apply, then the term does not give rise to a term in the succeeding 
column; the operations will at length terminate, and a complete scries of columns be 
obtained. Thus, if the letters are {a, b, c, d), and the operations are (as before) 
performed upon the entire .series of columns is 
a* 
d'b 
a^c 
a?‘d 
cdbd 
a^cd 
a'd~ 
abdr 
acd~ 
ad^ 
bd^ 
cd^ 
c‘‘ 
a-}? 
a'bc 
eder 
aV^d 
abed 
ae^d 
b'^d‘ 
bed" 
c^d' 
aV- 
oJJ^c 
ah(? 
ac^ 
b^cd 
bdd 
c\l 
b* 
b^c 
b^d 
bc^ 
c* 
MDCCCLXI. G 
