﻿454 Prof. R. Clausius on the new Conception of 



these changes cannot be regarded as correct consequences of 

 the earlier equations. 



I believe that Riemann subsequently convinced himself of this 

 error, and that this was the reason why he withdrew his paper. 

 Yet its publication has not been without use for science; for 

 though it has not yet solved the question as to the origin of elec- 

 trodynamic forces, it has contributed to moot this question afresh, 

 and to give to it a heightened interest ; so that at present several 

 physicists and mathematicians have had their attention directed 

 to it, and a solution may perhaps thereby be attained. 



Of the above-mentioned three papers, there remains to be con- 

 sidered that of Betti, which appeared in the Nuovo Cimento, 

 vol. xxvii. 



Betti figures to himself as follows the two constant currents 

 whose potential on each other he wishes to determine. He sup- 

 poses the closed curves traversed by the currents to consist of 

 elements which are periodically polarized, and therefore act upon 

 each other as if they were magnetic elements whose axes are pa- 

 rallel to the tangents of the curves. He assumes that the periods 

 of the changes are the same in both currents, so that they can 

 only be distinguished by their phases. He adds to this idea the 

 assumption that, for the propagation of the action of an element 

 of one current to an element of another, a certain time is neces- 



sary which he calls -, r being the distance of the two elements 



from each other. He thus obtains for the potential of the two 

 currents upon each other the same expression as that known in 

 electrodynamics. 



It might seem that the assumptions made had been fully con- 

 firmed, and that thus the question as to the nature of the electro- 

 dynamic forces had been solved in a different manner from that 

 attempted by Riemann and Neumann. Yet it may be shown 

 that here, apart from an improbability inherent in the mode of 

 representation, upon which I will not enter, there is an oversight 

 in the mathematical development which is of essential influence 

 upon the result. 



To show this, it is not necessary to follow the entire investi- 

 gation, but it is sufficient to consider that part in which it occurs. 



In order to express mathematically the periodical changes of 

 the polarized elements, Betti introduces a function of the time 

 <f>(t) which has the property of regularly altering its value in 

 very short intervals. The duration of the periods he designates 

 by p. Of this function the value is to be determined which it 



T 



assumes if t is replaced by t + a— -,in which a may be any mag- 



r c 



nitude between and p, and - the above-mentioned time which 



1 c 



