( 125 ) 



other words the force in the direction of V l>y a couple of rotation 

 ^^-^,5^. So we can regard tf^ as the foi'ce in the direction of V by 

 an isolated rotation in S. So that we must take as fictitious "force 

 field of an element of rotation unity" 



coth r, 

 directed perpendicularly to the radius vector. In reality, however, this 

 force field has rotation everywhere in Sp^. 



YII. Let us now find the scalar value U, function of /', which we 

 must assign to a planivector potential, tliat the "field of an element 

 of rotation unity" be its second derivative. We must have: 



dU _ 

 dr 

 U =z I cosech r. 



And we find for an arbitrary 2^' 



—^ I cosech r dr, 



1^ 



^^FMdx (//) 



And an arbitrary vector field X is the total derivative of the potential 



VIII. We may now^ wonder that here in iSp, we do not find 

 F^ and F^ to be identical, as the two derivatives and the two 

 potentials of a vectordistribution are perfectly dually related to each 

 other in the hyperbolic Sp^ as well as in the Euclidean ^S^^,. The 

 difference, however, is in the principle of the field property, which 

 postulates a vanishing at infinity for the scalar potential, not for the 

 planivector potential; and from the preceding the latter appears 

 not to vanish, so with the postulation of the field property the duality 

 is broken. 



But on the other hand that postulation in >$/>, lacks the reasonable 

 basis which it possesses in spaces of more dimensions. For, when 

 putting it we remember the condition that the total energy of a 

 field may not become infinite. As soon as we have in the infinity 

 of S^n forces of order g— '", this furnishes in a spherical layer with 

 thickness dr and infinite radius described round the origin as centre an 

 energy of order e— 2'- X «^"~"^^'' dr = e("— s)»- dr ; which for n ]> 3 would 



