68 Transactions of the Academy of Science of St. Lows 
An important theorem concerning uniform arcs is the follow- 
ing. 
T.15. If [4.2 is a uniform arc it is pessible to subdivide it so 
that if a; and a;,; are any two successive points of the net, then 
Wa,a;.,6forany6>0. This is the well-known Borel property. 
Integrability-R. Let f(x) be bounded in a region R, in which 
‘there is defined a uniform arc T',,. Let a system of nets be ap- 
plied to the arc and consider the sum S,, where 
PEL 
= >) f(x!) (ai — x), (A) 
and where x,;, x;41 are points of a net of the system, such that 
xo=a and x41=6, and x;Sx/ Sxji11. If the sequence ce has 
a limit S that is independent of the system of nets applied, 
f(x) is said to be integrable-R along [',,, and S, which is then 
called the line or curvilinear integral is denoted by 
. ee 
By the oscillation of f(x) between two points d; and di: of 
I’,, we shall mean the least upper bound of || f(y) ~F(x)Ih tae 
d;Sy, xSdj,,. The function f(x) is said to be continuous on 
the curve at a point Xo if x9 is interior to an interval for which 
the oscillation of f(x) is arbitrarily small. When the maximum 
oscillation over the set of intervals of I,, in a system of nets 
approaches zero as the order of the net increases, and this is 
true for all systems, then f(x) is said to be uniformly continuous 
on [.,. These definitions are, of course, only special cases of 
continuity and uniform continuity as previously defined. 
We have the following theorems. 
T.15. A function which is continuous on a uniform arc is 
also uniformly continuous on the arc. 
T.16. A function which is continuous on a uniform arc is 
also integrable-R on the arc. 
The proof of this theorem does not follow that ordinarily 
given in the theory of the real variable, in defining an upper 
and lower Riemann integral, but follows, rather, a modified 
form which does not make use of these auxiliary integrals. 
T.17. The elementary properties of the curvilinear integral 
are expressed by the following equations: 
