146 
and we then obtain for the various potential-components (using 
rational units) the following differential equation 
Pp pa Ppa 1 WY. 
On? Oy” dz’ * òf 
(a= 2, y, 2, t). 
hk 2. oe ee 
In this equations 6;, 6,, 6, are the components of the electric 
current, o, is the density of the electricity. The motion of the charges 
being given, Or, Oy, Oz, 6; are known functions of «, y, z, t. The 
general solution of the differential equations (3) may then be put in 
the following form: 
Let «,, y,, z, be the coordinates of a point for which p„ is to be 
found. The distance of this point to a point (a, y, z) may be called 
r, so that 
r? == (er) + (y—y,)* + (z—z,)* - - - - . (A) 
The sign of r is not fixed by this relation, but we may leave it 
undetermined in the mean time. 
A unit vector with components 
ee 
will be represented by r. The direction of vr is evidently dependent 
upon the choice of the sign of r. Let F be an arbitrary closed sur- 
face which encloses the point ae Yo. Zo: A surface-element of /, 
considered as a vector directed outwards, will be denoted by d%. 
Using these symbols we may write the general integral of the diffe- 
_ rential equation (3) in the form *): 
1 i Ox | r Opa r 
Pale, = = — fe V —+ | d8| —grad pad —+-9 pe 200) 
An r r cr Ot or 6 
The surface-integral has to be extended over the closed surface 
F, the space-integral over the enclosed space V containing the 
5 
point «,, %, Zo. The index t— — is meant to indicate, that at the 
Cc 
Nt Ops 
right-hand side the quantities dz, pa, grad pa, Ps refer to the time 
Ot 
a 5 
t— —, which varies from point to point. The double sign on the right-hand 
€ 
I) A proof of equation (6) may be found in Finska Vetenskapssocietetens 
Forhandl. L. |. 1908—09, Afd. A. n°. 6. If the sign of 7 is not fixed beforehand, 
it is easily found that the right-hand side has the double sign. 
