( 258 ) 



sin «—V dr \ sin «— V 



cos (p rj 



''— V dr 

 cos (p '' " ^ ' 



sin "—Ir Sn—l sin "—V Sn—\ 



This field has as only divergency two double points, in Pj and 

 P3, of which equal poles correspond in the opposite points. The field 

 E^ is then obtained by adding to it the field («) multiplied by è. 

 We find on the half "sphere between P^ and B: 



n 

 1 COS ^ . . 



'r dr. 



)s <p r , 



— — I sm "■ 



- ' ■ Sn 1 sin 



On the half "sphere between P, and B: 



r 



1 costp r , 



." ' . — I S171 n-^^r dr. 



Sn—i sin^~^rj 

 



Or, if we define on both halves the coordinates r and y according to 

 P, and PjQi, we arrive at the expression holding for both halves: 



1 cos<p r , 



. . I sm "— 'r dr ■^;:^ ip„ \r .) cos (f. 



Sn—\ sin "— ^r J 

 r 



III. For the potential of the fictitious "field of a single agens 

 point" we find : 



ƒ• 



ipn {r) dr — P, (r). 

 And for the arbitrary gradient distribution holds 



lx^y/J\^FAr)dr (/) 



Of the divergency distribution of F^{r) in points of a general posi- 

 tion we know that, taken for two completely arbitrary centra 

 (fictitious agens points) with opposite sign and then summed up, 

 it furnishes : so on one side that distribution is independent of 

 the position of the centre and on the other side it lies geome- 

 trically equivalent with respect to all points ; so it is a constant. 

 But if the function F^ (r) has constant divergency in points of general 

 position it satisfies a differential equation putting the divergency 

 constant. In this is therefore a second means to determine the func- 

 tion Pi and out of this the field P^. 



The differential equation becomes : 



