94 
MATHEMATICS: G. C. EVANS 
Proc. N. A. S. 
F(si) and F(s 2 ) ; and if a point P is a vertex for si, a portion of the possible 
discontinuity at P proportional to the angle at the vertex is taken up by 
F(si). 
In order to illustrate this correspondence, take the following example. 
Let 2 be the square 0<x<l, 0<^<1; and to the points M 1 of rational 
coordinates in this 2 assign positive numbers p(M*) in such a way that 
when the points are arranged in countable order, the series of numbers 
will be convergent; and make the definition 
M = f{e) = S^(M'), (16) 
where the sum is extended over all the M* in the set e. Introduce now 
an auxiliary function n(0,M), which for a given point M depends on an 
angle 0 made with a fixed direction, and which is such that the quantity 
J2ir 
fjL(6,M)d$ is everywhere unity. Consider the quantity 
H(M,e h 6 2 ) = Cfi(0,M)dd. 
J di 
For the purpose of defining F(s) every point of s may be considered 
temporarily as a vertex, the tangents forward and backward from M 
making, respectively , angles 0i,0 2 with the fixed direction. Then the equa- 
tion 
F(s) = S>(M f ) + 2 s H(M i ,0 1 ,0t)p(M i ), 
where the first summation takes all the points M* interior to <r and the 
second takes all the points M l on the boundary s, satisfies the equation 
of correspondence (14) and defines an additive function of curves of limited 
variation which is discontinuous on every curve which contains a rational 
point. In general, the discontinuities of F(s) will not be of the first kind; 
in order to make them so, and thus satisfy the equation (15), it is sufficient 
to put p(0,M) = 1/2tt. 
The directional derivative of a point function u(M) is not necessarily 
a vector, merely on the basis of its existence. If, however, this derivative 
for two directions, say bu/dx and bu/dy, is summable superficially and 
satisfies for these two directions and all curves of T the equation (13), 
then the vector which has these two quantities for its components in the 
two directions will be a gradient vector, and the original function its poten- 
tial function. Yet it is simpler for the purpose of proving theorems and less 
artificial, to use directly a generalized partial derivative which has itself 
vectorial properties. 
We say that D a u, the generalized derivative in the direction a of u(M), is 
the limit, if such limit exists, of the expression 
D a u = — - f uda' (17) 
(7 = 0 a J s 
where the fixed direction a' makes an angle of ir 1 2 with the fixed direction 
a, and a denotes the area enclosed by s; it is understood that a tends 
