Outline of a Theory of Functions of an Abstract Variable 67 
h'(x) = f(x) + g’(x). 
T.12. If f(x) is differentiable, and ais a scalar, then 
taf(x)}’ = af"(2). 
T.13. If f(x) and g(x) are differentiable, and if 
h(x) = f(x): g(x) 
then 
h'(x) = f(x)-g'(x) + f'(x)-g(a), 
Provided, of course, that the factors are multiplicable. 
T.14. If the derivatives of f(x) and g(x) exist and are multi- 
plicable, and if z - f(y), where y =g(x), then 
ag: ae ay 
dx dy dx 
As an illustration, we remark that the derivative of @,)x” is 
simply Y-@{n\)x""', where r is, of course, a positive integer. 
Curves. We shall be concerned with the class of linear con- 
tinua that are described as rectifiable Jordan curves without 
multiple points, which, for brevity, we shall call uniform curves. 
There exists a relation of order between the points of such 
Curves, this order relation being denoted by the customary signs 
S,and >. A uniform curve I’, having a first point a@ and a 
last b, and containing these points is termed the uniform arc 
P, and its length is denoted by Yas. This length is deter- 
mined by means of a system of nets defined as follows: Let 
@,@2,---,a,ben arbitrary points on I’,», satisfying the con- 
dition. that 
a<a<ae< + + -<On<. 
Such a subdivision is called a net of order m, and an infinite se- 
quence of nets corresponding to the sequence of positive in- 
tegers {n} is called a system when between any two points of a 
net there are points which belong to every net of order m >some 
No. 
By the oscillation w,» of a uniform curve I’ between two of its 
Points a and b, we shall mean the least upper bound of l|x—y| 
where the range of x and y is the arc Tas. 
