On Differences and Differentials of Functions of Zero. By WILLIAM R. 
HAMILTON, Royal Astronomer of Ireland. 
Read June 13, 1831. 
Tue first important researches onthe differences of powers of zero, appear to be 
those which Dr. Brinxtey published in the Philosophical Transactions for the year 
1807. The subject was resumed by Mr. Herscuer in the Philosophical Transactions 
for 1816; and in a collection of Examples on the Calculus of Finite Differences, 
published a few years afterwards at Cambridge. In the latter work, a remarkable 
theorem is given, for the development of any function of a neperian exponential, by 
means of differences of powers of zero. In meditating upon this theorem of Mr. 
Herscuet, I have been-led to one more general, which is now submitted to the 
Academy. It contains three arbitrary functions, by making one of which a power 
and another a neperian exponential, the theorem of Mr. Herscuet may be obtained. 
Mr. Herscuer’s Theorem is the following : 
S(EJ=fAttf tA) o + 5 FU+A)e +&e. (A) 
JF (1 +4) denoting any function which admits of being developed according to posi- 
tive integer powers of A, and every product of the form A™ o” being interpreted, 
asin Dr. Brinkiey’s notation, as a difference of a power of zero. 
The theorem which I offer as a more general one may be thus written : 
¢A+4) f¥O)=fA+4) oA+4) (0) "5 (B) 
or thus 
F(D) ft (V=fL+4) FD) (0) )”. (C) 
In these equations, f, ¢, F, y, are arbitrary functions, such however that f(1 +A’), 
¢(1 +A), (D), can be developed according to positive integer powers of A’ A D ; 
and after this development, A’ A are considered as marks of differencing, referred to 
the variables o’ 0, which vanish after the operations, and D as a mark of derivation 
by differentials, referred to the variable 0 . And if in the form ( C) we particularise 
VOL, XVII. 3D 
