66 Transactions of the Academy of Science of St. Louis 
isnotanullifyingelement. It is natural that the binomial theo- 
rem should hold also for abstract polynomials. 
10. 
Z r 
Ain(% + y)" = a4 af yea . 
0 
The derivative. Let y=f(x) be defined for a region R, and 
let it be continuous at the point x=x . Then f(x) is said to 
have a differential at x, if there exists a function f’(x»)dx, linear 
continuous homogeneous in dx, such that for 6>0 
[f(a0 + dx) — f(x) — f'(xo)dx|| < 4||dx| 
when ||dx|| is sufficiently small. We write 
dy = f'(x)dx, 
and dy is called the differential of f(x) at x, provided, of course, 
that it exists. 
It may be seen from the definition that dy is an element in 
the same spaceasy. If this space is E,,), then according to T.7 
f'(x) is an element of E,,41;. This element is called the deriva- 
tive and will be denoted sometimes by dy/dx. Obviously the 
derivative is the analogue of the derivative in the theory of the 
complex variable, where there as well as here, the essential fea- 
ture is that the “direction” of dx as ||dx|| tends to zero is imma- 
terial. Thus, as is well known, if the differential exists, it may 
be obtained as 
d 
a + &dx) = 
for any choice of dx. 
If f(x) is differentiable at x we may write 
fl + dx) = fla) + f(a)dx + |[dal] -e(2, dx), 
where ||e(x, dx)|| <6 for ||dx|| sufficiently small. In general, 4 
will depend upon x. When, however, it does not, we shall say 
that f(x) is uniformly differentiable in the given range. 
The following theorems are elementary. 
T.11. If f(x) and g(x) are differentiable, and if 
h(x) = f(x) + g(x) 
then 
