Voi,. 7, 1921 
MATHEMATICS: G. C. EVANS 
91 
jv n Uds = F{s) 
jv n Vds = G{s), 
which may be written in the form given by (5) or (5') plus harmonic 
functions (i. e., solutions of (1') after the unnecessary discontinuities 
have been removed). With respect to F(s) and G(s) let us introduce the 
symbols T$(s) and T G (s) to stand for their total variation functions. 
Then we have the following rather general result. 
Theorem II. — If the discontinuities of F(s) and G(s) are of the first 
kind, a sufficient condition that the equation 
fa (V x UV x V + V y UV y V)d<r = f UdG(s)-f UV n Vds 
= f VdF(s)- f W n Uds (7) 
J <r J s 
shall hold for all regions a and all boundaries 5 of T, internal to 2', is 
the existence of the quantity 
f log J- dT F (s)dT c (s). 
J 2' r MiM 
A vector <p which satisfies the equation corresponding to (3') 
jjp n ds = F(s) (8) 
for every curve of T may be called a polarization vector for the distribution 
F(s) ; it is not necessary that such vectors shall, as in the examples already 
given, have potential functions. Those that we have already given satisfy 
also certain restrictions of integr ability which we denote by condition N, 
as follows : 
Condition N. Any component of <p is summable superficially, and the 
normal component is summable along any curve of T ; in particular, the 
integral of the absolute value of <p n , along any curve of T or finite number 
of mutually exclusive curves remains finite, less than some fixed number 
N, provided that the total length of the curves remains finite, less than 
some fixed number s 0 . 
We have then the following theorems : 
Theorem III. — Let G(s) be an arbitrary function of curves, additive 
and of limited variation, 7 and let \[/(M) be a polarization vector for it 
which satisfies condition N; further let u(M) be a function continuous 
over 2' with its first partial derivatives. Then for every 5 of class V 
the following equation is valid: 
fudG(s) = futJs + Jjg +, + g (9) 
Theorem IV. — If yf/{M) satisfies condition N and u{M) is continuous 
and a potential function for its vector gradient in 2, the equation 
