9A Mr Herschel on the Application of a new mode of Analysis 
another, from those employed to represent quantities themselves, 
and of expressing the relations subsisting between operations of 
different kinds, by equations among their characteristic letters, 
is due to M. Arbogast ; and however wide and interesting the 
field it lays open to speculation, I am not aware that it has been 
much cultivated by others. In a late publication I have endea- 
voured to shew the elementary simplicity and luminous elegance 
which the use of this method is calculated to introduce in the 
method of differences ; and in a paper printed in the Transac- 
tions of the Royal Society for 1816, have endeavoured to ex- 
tend the method itself to cases not within the contemplation of 
its inventor, and to make it the basis of what I think may fair- 
ly be termed a new mode of symbolic representation, and one 
possessing peculiar advantages where it can be employed. In 
the following short essay, I might perhaps take for granted the 
fundamental theorem there demonstrated ; but as the original 
proof is unnecessarily complicated, the following one, which is 
much shorter and more simple, will not be irrelevant. 
(1.) Let f(f) represent anyfunctionof e^? (where 71 828....) 
and let us suppose it developed in powers of t in the series, 
Ao + A^. + Ag. t^ -p &c. 
To determine the coefficient of we have 
/(/)=/{ ! + (/-!)} 
1 (A 
(1) + . f (1) + -r (1) + &c. 
Now, the co-efficient of t^ vofif) is y (!)• in the second 
term‘d Jf .f ( 1 ) it is . In the third, 
(/ — I)' 
1.2 
/" (1) or 
1.2 
r W it is- 
2^ — 2.P-hC^ 
1.2 
f" ( 1 ) 
X ^ and so on. But the numerators of these fractions are 
respectively the first, second, &c., differences of the powers 
of the natural numbers 0, 1, 2, &c., and denoting these by 
A0% &c., the whole co-efficient of f will be 
* Translation of Lacroix’s Difierential and Integi-al Calculus.-— Appendix. 
