( 25E ) 
fll. If w and g are functions of «, y and z, w satisfying the 
three following conditions : 
Ist w and its first derivatives with respect to 2, y and z are 
continuous everywhere. 
Qnd some isolated points, lines and surfaces excepted, w fulfils 
in an acyclic region, the equation: 
Vyp=Pw—4nag. 
d d du 
3rd the products zw, yw, ew, we : p= and oe become nowhere 
at u a 
infinite; then the potential of an agens, the density of which is g, 
is for that region w, the potential function being: 
Ae-or 
Pp (r) = 
In order to prove this theorem, we take into account that the 
. : : ; Bue Ae 
potential of an agens, for which the potential function is : 
7 
fulfils the differential equation : 
ws yr Be AG TR Ske AD) 
which is a special case of equation (3). Íf we can prove that on 
the given conditions only one solution of (11) is possible, the theorem 
is proved. We shall do so by proving that if there are two solu- 
tions, the difference of these functions will be zero everywhere. 
If w and v are two solutions of equation (11) and if we put 
y—v=u, the new function u will satisfy the equation: 
v2 ie q° Us 
As w and v and their first derivatives with respect to «, y and 2 
are supposed to be continuous everywhere, this is also the case 
with the function u and we may make use for this quantity of the 
well-known theorem of GREEN. This furnishes the equation : 
du du = du 
(Wu) ude == U Hes COS & + — cos [9 -\- zer) da == 
de dy dz 
NE & rl (@ Sik dur?) : ie 
a) +(¢) | (i) av. ° ( ) 
