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MATHEMATICS: G. C. EVANS Proc. N. A. S. 
(10) 
f udG(s) = r ^ w J5 + f (V^ + VyU^da (90 
remains valid provided that one of the following hypotheses (a), or 
(7) be imposed: 
(a) Vu(M) is bounded. 
(0) is bounded. 
(7) The quantities {Vu(M)\ 2 and {^(M)} 2 are summable superficially. 
If in this theorem we put u(M) = log 1/r, we obtain: 
Theorem V. — If ^(M) satisfies condition N the equation 
f log - dG{s) = f log I * n ds + 
/„{^ ,og r 1 + ^| log ;K 
is valid, given the region tr, for all points Mi except possibly those which 
form a point set of superficial measure zero. 
One or two more special theorems are perhaps interesting. If in Theorem 
II we put V =log 1/r, we obtain the equation 
2*t/(M 1 ) = j I U ~ logi -log i V,!*!*? + J log i dF(5) (11) 
which holds whenever C/(Mi) is the derivative of its own superficial 
integral, and, therefore, except for the points of a set of superficial measure 
zero. 
If in Theorem IV we put 
where X is a scalar point function, so that the vector (p satisfies the equation 
\(p n ds — G(s) 
s. 
then equation (9 r ) reduces to the following: 
/\{\ 7 x u<p x + VyU(p y }d<7 — — I \uip n da +J udG(s). (12) 
a J f J o 
These theorems provide for the Stieltjes integral with respect to a 
function of curves a representation in terms of a Lebesgue integral and 
a curvilinear integral. The curvilinear integral is essentially a curvilinear 
integral, depending only on the contour, and differs thus from the integral 
around general boundaries defined by P. J. Daniell, 9 which is a "frontier 
integral" not uniquely determined by the contour itself, but by some super- 
ficial point set. 
The theorems of this section remain true if for a is substituted a set of 
points measurable superficially in the sense of Borel, for 5 its frontier, 
and for F(s), G(s) the corresponding additive functions of point sets 
f( e )> g( e )> the frontier integrals being defined according to the method of 
Daniell. We shall come later to another possible method of determining 
frontier integrals. 
