96 
MATHEMATICS: G. C. EVANS 
Proc. N. A. S. 
with 
v(Md = I C?-2 dm + zrp dm- 
From this evaluation follows the equation 
W(z 1 )dz 1 = F( Sl ), (21) 
51 
where F(s) is the function of curves with discontinuities of the first kind 
which is equivalent to f(e) . 
If we extend the notion of generalized derivatives to functions of a 
complex variable, it turns out that what corresponds to the quantity 
dx dy 
is the limit: 
i^If W(z)dz, 
a = 0 la J 
which is merely the superficial derivative, where it exists, of the additive 
function of point sets f(e) /i. But this quantity exists in S except possibly 
at the points of a set of superficial measure zero. At a point where this 
superficial derivative itself vanishes it seems opportune to follow the di- 
rection of Borel's concept, and speak of the function W{z) as monogenic. 
If k(z) is a function of a complex variable, analytic in 2, and Q(z) is 
a solution of the equation 
Q(z)dz = F(s) 
S 
and equal to W(z) plus an analytic function, the equation 
f K Qdz = f K dF(s) 
J s , J <J 
is satisfied. In particular, by taking as k(z) the function l/(z—Zi), it 
may be deduced that except possibly at the points of a set of superficial 
measure zero the following identity holds: 
o(*o = 9 -U f m dz +'( <m (22 ) 
ZTt J s z — Z\ Ziri J a z — Z\ 
The function given by (20) is the unique derivative of a certain func- 
tional of open curves in the plane, considered as a function of the end- 
point of the curve. We have, in fact, W(z) = dZ(l\z)/dz, where 
£(l|zi) = gij" log (z-zjdfie). (23) 
Here the values of arctan y— yi/x— %i are taken, say, as the principal values 
for a particular value of 01 = zo, thus defining log (0—20) for z in 2; log 
(z—Zi) is then defined in general by means of the curve 1 which connects 
Zq with Zi. 
