Vol. 7, 1921 
MATHEMATICS: G. C. EVANS 
93 
3. Definitions and Examples. — It is desirable to render more precise 
some of the notions which we have used. 
A curve of class T is a simple closed rectifiable curve composed of a fiinite 
number of pieces, each piece having a definite direction at every point; 
for each curve there is a constant T such that the following inequality 
holds. 
f |coswr| rfs<r, 
wherever the point Mi may be. For a curve which is everywhere convex, 
for instance, T may be taken as 2t. 
With respect to gradient vector and potential function we have the fol 
lowing definition. If A(M) is a vector point function whose components 
in two fixed directions (and therefore in any fixed direction) are summable, 
and u{M) is a scalar point function whose integrals may be defined along 
any curve of class T, then A is spoken of as a gradient of u, and was a 
potential function of A provided the equation 
f A a da = f uda' (13) 
J a J s 
is satisfied for every 5 of T and for every a, the subscript a being used to 
represent a fixed direction and a' that direction advanced by it Ji- 
lt is sufficient that (13) should hold for two distinct directions a in 
order to hold for every direction. 
The curvilinear integral in (13) is an example of what may be called 
an absolutely continuous function of curves. For the purpose of this article 
we say that an additive function of curves is of limited variation if it is 
the difference of two additive functions of curves, each one positive or 
zero for every curve of class V. 
Every curve 5 of T determines a set e of points interior to the region a 
which it bounds. We shall consider correspondences between functions 
of curves F(s) and functions of points sets /(<?) such that for every curve 
on which F (s) is continuous the equation 
f(e) = F{s) (14) 
holds. Given F{s) it may be shown that/(^) is uniquely determined over 
all sets measurable in the sense of Borel, simply by means of (14). 
On the other hand the equation (14) is not sufficient to determine F(s), 
given f(e). The discontinuities of F(s) may still be distributed in various 
ways. The F(s) which is defined by the formula 
F(si) = ~ f dsA df(e) (15) 
has all of its discontinuities of the first kind; that is, if a portion, s', which 
has everywhere a tangent, forms part both of Si and s 2 , otherwise mutually 
exterior, a possible discontinuity upon s' would be shared equally by 
