236 On Differences and Differentials of Functions of Zero. 
the functions #’, ¥, by making F’ a power, and ya neperian exponential, we deduce 
the following corollory : 
TEGCO=af AS A )Be =fK$ CEPA) 0" ; 
A : e 
that is, the coefficient of 
1.2... 2 
in the development of f(e') may be represented by 
Ff (1 +A) o* ; which is the theorem (4) of Mr. Herscuet. 
June 13, 1831. 
ADDITION. 
The two forms (B) ( C) may be included in the following : 
VF¥O) =fA+AV (eC) Y. (D) 
To explain and prove this equation, I observe that in MacLaurin’s series, 
a PFO). 2 Df(0) 
fn (Gt) = 5(O) et oat eps ae 
we may put « =(1+A) 2° and therefore may put the series itself under the form 
S(e)=f(0) +72. 1+ ayer +P. 
or more concisely thus 
(1+A)’ x +&c. 
Fi (a) = fA): (E) 
which latter expression is true even when Mactaurin’s series fails, and which 
gives, by considering » as a function ¥ of a new variable o’ and performing any 
operation vy’ with reference to the latter variable, 
Vt (CO) =VFA+4) HO) Y- (F) 
If now the operation y' consist in any combination of differencings and differ- 
entiatings, as in the equations (B)and (C), and generally if we may transpose the 
symbols of operation vy’ and f(1 +A), which happens for an infinite variety of forms 
of VY’, we obtain the theorem (D). It is evident that this theorem may be extended 
to functions of several variables. 
June 20, 1831. 
eos 
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