457 
II. If we take the differential coefficient of x", and mul- 
tiply it by «, the result will be 2"; that is to say, 
et < ns 
ona a"; 
and as a consequence of this equation, we shall likewise have, 
a" ¥(n) = F (of) 2", (2) 
In the right-hand member of this equation, let us put 
1 +x —1 in place of x; and then expand by the binomial 
theorem ; the result will be 
d d 
u"K(n) =r(27) 24 Feaerd. y e az) foe ie —1l) 
dx l 1.2 
+ &e. (3) 
The coefficient of n (n—1)...(n—m-+1) in this deve- 
lopment will be 
: “ataersite HN of pha 2)—&e 
1.2 
and, if we now suppose x = |, we shall have the development 
of F (z) in the desired form ; the coefficient of the factorial 
1.2..m {en (m) —ma"—"¥(m—1)+ 
n(n—1)..... (n—m-+ 1) being 
at! Uae el) ——) F (m—2)— &e.} 
Comparing the two expressions (1) and (3) we find, as we 
ought to do, 
> m(m —1) 
Faas oi alg F(m—2)— &e., 
-# ° 
_ a formula which might be obtained directly by making «=0 
in the fundamental equation of the calculus of finite differences, 
A™r (0) = F(m) —mrF (m—1) + 
m (m —1) 
1.2 Ur4m—2 et &e. 
A” ee = Urym— MU syn 
