( 73 ) 



(]eri\ative. in otlier words eacli vectordistribiitioji has a potential 



and thai i)<iifMili;il i- uiiifnrinlv determined liv it. 



Proof. Let T" be the fuiven distribution, then 



is its potential. For \7^P=x7F, or y(^P) = vF, or \jP=V. 

 Farther follows out of the field property of V, that P is uniformly 

 determined as \7~- of v I'^, so as V of V. So P has clearly the 

 potential property: it holm I, however, not have the field property. 



X.B. A distrihiition not to be regarded here, because it has not 

 the field profierty, thougii it has the potential property, is e. g. the 

 tic-tit ions force field of a single agens point in aS,. For, here we 

 have not a potential vanishing at infinity — and as such deter- 

 mined uniformly. The magnetic field in ^S', has field property and 

 also all the fields of a single agens point in S^ and higher spaces. 



Let us call "'' T the first derivative of /' F and TyFthe second; 

 we can then break up /' V into 



and 



From the preceding follows immediately : 



p II— \ p 



Theorem 7. Each /,_i T has as potential a j, V. Each p_|_i V has 



as potential a p I . 



V 



We can indicate of the ^;_^i I' the elementary di^>trilultion, i. e. that 



p 

 particular ^j+i V of which the arbitrary Sn integral must be taken to 



i' 

 obtain the most general /;-ri F. 



For, the general p^\V is "^ of the general ''"*" F, so it is the 

 general >S„ integral of the ^ of an isolated {p -f l)-dimensionai 

 vector, which, as is easily seen geometrically, consists of equal /'vectors 

 in the surface of a /'s[)here with infinitesimal radius described round 

 the point of the given isolated vector in the /i'^,-|_i of the vector. 



V 



111 like manner the general ^.-i F is the ^y of the general P-^ I , 



