Vol. 7, 1921 
MATHEMATICS: G. C. EVANS 
95 
toward 0 only in such a way that the ratio ay d 2 , where d is the diameter 
of cr, remains different from zero by some positive quantity. 
It is a theorem that the expression (6) is the generalized derivative 
of the expression (5) except at the points of a set E 0 of superficial measure 
zero, which may be chosen independently of the direction a. The function 
defined by (5) is therefore a solution of the equation 
D n u ds = F(s) (18) 
for any curve which does not contain points of E Q of more than linear 
measure zero. The transition from this equation to the equation 
S On 
involving derivatives in the customary sense, is possible for classes of 
curves defined in terms of curvilinear coordinates, by means of the identity 
between a double and an iterated integral in these coordinates. 
4. A Double Integration by Parts. — The Green's theorem from the point 
of view of section 2 may be regarded as an integration by parts of the 
multiple Stieltjes integral with respect to one dimension only. It is 
perhaps worth while to state the theorem which one obtains by a com- 
plete integration by parts of the double integral. 
Let u(M) be a continuous point function of limited variation 10 and let 
q(M) — G(s) where 5 is the rectangle whose diagonally opposite vertices 
are M and the origin. Then if 5 is any simple closed regular curve, and 
points Pi with coordinates x^yi are taken round the curve such that 
I — %i y%+\ ~" y% J<5 the following integrals have meaning: 
/ 
ud y q = hm EM%i>m){q(xiiyi±i)--<i(%i,yi) 
0=0 
ud x q = hm 2ju($i,yi){q(x i+ uyd-q(x it yi) 
0 = 0 
yi>miy i+ i, (19) 
\q(%i+i,yi)-q 
and the identity 
/udq = I qdu + I dqu — I ud y q + I ud x q (19 ; ) 
is valid. 11 In this equation the quantity J* dqu stands for the total varia- 
tion of qu over the region a. 
5. Applications. — Consider the function 
p. f d M (20) 
where z denotes the quantity x + iy, where f(e) — fi(e) + if 2(e) is a 
complex additive function of point sets, and the integral has the meaning : 
W(z0 = U(Mi) + iV(Mi) 
