( 127 ) 



The same holds for the arbitrary o^; of (he lines of force one 

 part goes to infinity ; the potential lines however are closed. 



X. If we now have to lind the field with rotation only, giving 

 for given rotation distribution a niinininni enei'gy, it follows from a 

 consideration of the rotation as divergency of the normal vector, that the 

 scalar value of the planivector potential at infinity must be 0, and the 



general 2^ is composed of fields E„, derived from a planivector 



SZTl (£) 



potential —_ (whilst we found under the postulation of the field 



sink r 



property sin (f- coth r). 



In contrast to higher hyperbolical spaces and to any Euclidean 



and elliptic spaces the fields E^ and E^ cannot be summed up here 



to a single isolated vector. 



For this field E^ and likewise for the arbitrary 9 A' the lines of 

 force (at the same time planivector potential lines) are closed curves. 



XI. We have now found 



Jx = xv/ 



2jr 

 1 



I coth I r dx, 



1 r \ï7 2X 



2X = \2/ I Az I coth \ r dr. 



And from this ensues that also the general vector distribution X 

 having under given rotation and divergency a minimum energy is 

 equal to : 



rwA . rwA 



Xdiv. + Xrot. = \V I ~— I coth ^ r dx + \y I -^ I coth \ r dx. 



For, if V is an arbitrary distribution without divergency and without 

 rotation in finite, it is derived from a scalar potential function, so it 

 has (according to § VIII) no reciprocal energy with Xdiv.', neither 

 (as according to § IX all lines of force of Xrot. are closed curves 

 and a flux of exclusively closed vector tubes has no reciprocal 

 energy with a gradient distribution) with Xrot. ; so that the energy 

 of Xdiv. + Xrot. + F is larger than that of Xdiv. + Xrot. • 



So finally we have for the general vector distribution of minimum 

 energy X: 



'J 



Z^A 

 A = V I . I coth I r dx. 



2n 



