( 72 ) 



The terms under the sign ^ JS" of Tj are annulled hv those of 

 1\, so (hat only 



h = n 



i: 



h — \ 



is left. 



Corollary. If a veetordistribntion v V is given, then the vector- 



r Vdv 



distribution I -— „, integrated over the entire space, has tor 



J /;„(» — 2)r"-- 



second derivative T^. (if /,„/•''"' expresses the surface of the 



"-'-sphere in ;S;,). 



The theorem also holds for a distribution of sums of vectors of 

 various numbers of dimensions, e.g. quaternions. 



We shall say that a vectordistribution has the potential property 

 when its scalar values satisfy the demands of vanishing at infinity, 

 which must be put to a scalar potential function in Sn- ^) And in 

 the following we shall suppose that the vectordistribution from 

 which we start possesses the potential property. Then holds good: 



Theorem 4. A vectordistribution V is determined by its total 

 derivative of the second order. 



For, each of the scalar values of V is uniformly determined by 

 the scalar values of yM", from which it is derived by the operation 

 dv 



J 



X-„(/.-2)r"-2 



Theorem 5. A vectordistribution is determined unifoi'mly by its 

 total derivative of the first order. 



For, from the first total derivative follows the second, from which 

 according to the preceding theorem the vector itself. 



We shall say that a vectordistribution has the pelil property, if 

 the scalar \alues of the total derivative of the first order satisfy the 

 demands which must be put to an agens distribution of a scalar 

 potential function in S,r And in the following we shall suppose 

 that the vectordistribution under consideration possesses the field 

 property. Then we have: 



Theorem 6. Each vectordistribution is to be regarded as a total 



1) Generally the condition is put : the function must become infinitesimal of 

 order ?i— 2 willi respect to the reciprocal value of the distance from the origin. 

 We can, however, prove, lliat the being infinitesimal only is sulUcienl. 



