( 118 ) 



It is the tirst derivative of a scalar distribution : 



— 1 -\- coth r, 

 and has in the origin an isolated divergency of 4:jr. 



V. In future we shall suppose that X has the field property and 

 shall understand by it, that it vanishes at infinity in such a manner 

 that in the direction of the radius vector it becomes of lower order 



than - and in the direction perpendicular to the radius vector of 

 r 



lower order than e"^ . 



For a 0-^ this means that it is derived from a scalar distribution, 

 having the potential property, i.e. the property of vanishing at infinity. 



Now the theorem of Green holds for two scalar distributions (corap. 

 Fresdorf, diss. Göttingen, 1873): 



r oil? r C ^^ r. C 



tip — dO — I ^ y " ip . cZt =: \y^ — dO — I \p ^^(p . dx 



j = I aS \grad. <p, grad. i^j dt. j . 



If now (f and if? both vanish at infinity whilst at the same time 

 Urn. (p\l^ e-'' ^ 0, then the surface integrals disappear, when we apply 

 the tiieorem of Green to a sphere with infinite radius and 



i (f . s^"" \p . dr z=z j If? . y" ^ . cZt, 



integrated over the whole space, is left. 



Let us now take an arbitrary potential function for (p and 

 — \ -{- coth r for tf% where 7' represents the distance to a point P 

 taken arbitrarily, then these functions will satisfy the conditions of 

 vanishing at infinity and /m. (p if? e-'' = 0, so that we find : 



Ajt . (f p := I ( — 1 -|- cof/i r) \y'^(p . dr. 



So, if we put — 1 + (^oth r ^ Fiir), we have : 



1.. _r\27o-x 



F,(r)dT (/) 



VI. We now see that there is no vector distribution with the field 

 property, which has in finite nowhere rotation and nov/here diver- 

 gency. For, such a vector distribution would have to have a potential, 

 having nowiierc rotation, but that potential would have to be every- 

 where according to the formula, so also its derived vector. 



