( 120 ) 



whilst the condition for H is: 



d (Ildh) = dh ds X sin f. 



So we have but to take — for H. 



'dh 



To find -2" we integrate the current of force within the meridian zone 



through the spherical surface through y-*. The force component perpen- 



coshr 

 dicular to that spherical surface is 2 cos tp —_ — — , therefore : 



sink r 



f 



/coshr 

 2 cos (p — ; . siiihr d'p . sinhr sin (f dO- = dd- coth r . sin^ (p. 

 sinker 







So: 



2 ^ coshr 



H ■=: — :=. — ; : = — sm (f. 



dh sinhr sin (fdd- sinh^r 



X. From this ensues, that if two arbitrary vectors of strength unity are 



given in different points along whose connecting line we apply a third 



coshr 



vector = , the volume product of these three vectors, i. e. the 



sinh'^r 



volume of the parallelepipedon having these vectors as edges taken 



with proper sign, represents the linevector potential according to the 



first (second) vector, caused by an elementary magnet with moment 



unity according to the second (first) vector. 



To find that volume product, we have first to transfer the two 

 given vectors to a selfsame point of their connecting line, each 

 one parallel to itself, i. e. in the plane which it determines with that 

 connecting line, along which the transference is done, and maintaining 

 the same angle with that connecting line. 



The volume product \\i{S^,S^) is a symmetric function of the two 

 vectors \m\ij of which we know that with integration oï S^ along 

 a closed curve s^ it represents the current of force of a magnet unity 

 according to -S^ through s^, in other words the negative reciprocal energy 

 of a magnet unity in the direction of S^ and a magnetic scale with 

 intensity unity within s^, in other words the force in the direction 

 of aS, by a magnetic scale with intensity unity within s^, in other 

 words the force in the direction of S^ by a current with intensity 

 unity along «j. So we can regard V^{S^,S^) as a force in the direction 

 of S^ by an element of current unity in the direction of S^. 



With this we have found for the force of an element of current 

 with intensity unity in the origin in tlio direction of the axis of the 

 system of coordinates : 



