( 76 ) 



along the circles, which project themselves on the plane of Vo, 

 as circles concentric to Vo-, and where <p is the aiigle of the radius- 

 vector with the normal on the circular current. The field E^ is the 

 second derivative of the plan i vector potential, i.e. the rotation of 

 the normal linevector. 



According to what was derived before the field E^ of a small 

 circular current is outside the origin equal to the field E^ of an 

 elementary magnet normal to the current. 



In this way we have deduced the principle that an arbitrary 

 force field can be regarded as generated by elementary magnets and 

 elementary circuits. A finite continuous agglomeration of elementary 

 magnets furnishes a system of finite magnets; a finite continuous 

 agglomeration of elementary circuits furnishes a system of finite 

 closed currents, i.e. of finite dimensions; the linear length of the 

 separate currents may be infinite. 



Of course according to theorem 6 we can also construct the 



/l 

 scalar potential out of that of single agens points I — X l^ie second 



derivative of the field), and the vector potential out of that of rectilinear 



1 

 elements of current (perpendicular to -— X the first derivative of the 



field), but the fictitious ''field of a rectilinear element of current" has 

 everywhere rotation, so it is the real field of a rather complicated 

 distribution of current. A field having as its only current a rectilinear 

 element of current, is not only physically but also mathematic- 

 ally impossible. A field of a single agens point though physically 

 perhaps equally impossible, is mathematically just possible in the 

 Euclidean space in consequence of its infinite dimensions, as the 

 field of a magnet of which one pole is removed at infinite distance. 

 In hyperbolic space also the field of a single agens point is 

 possible for the same reason, but in elliptic and in spherical space 

 being finite it has become as impossible as the field of a rectilinear 

 element of current. The way in which Schering (GiHtinger Nachr. 

 1870, 1873; compare also Fresdorf Diss. Gottingen 1873; Opitz 

 Diss. Gottingen 1881) and Killing (Orelle's Journ. 1885) construct 

 the potential of elliptic s[)ace, starting from the suj^position tliat 

 as unity of fiehl must be possible the field of a single agens point, 

 leads to absurd consequences, to which Klein (Vorlesungen iiber 

 Nicht-Euldidische Geometrie) has referred, without, however, proposing 

 an improvement. To construct the potential of the elliptic and 

 si)hcrical spaces nolhing but the field of a double point must be 

 assumed as unity of field, which would lead us too far in this 



