Middle of the Nineteenth Century. 261 



It will be seen that Kirchhoff's electrical researches were 

 greatly influenced by those of Weber. The latter investiga- 

 tions, however, did not enjoy unquestioned authority ; for there 

 was still a question as to whether the expressions given by 

 Weber for the mutual energy of two current elements, and for 

 the mutual energy of two electrons, were to be preferred to the 

 rival formulae of Neumann and Eiemann. The matter was 

 examined in 1870 by Helmholtz, in a series of memoirs* to 

 which reference has already been made.f Helmholtz remarked 

 that, for two elements ds, ds', carrying currents i, i', the electro- 

 dynamic energy is 



n'(ds.ds') 

 r ' 

 according to Neumann, and 



?V 

 5-(r.ds)(r.ds'), 



according to Weber; and that these expressions differ from 

 each other only by the quantity 



- cos (ds . ds') + cos (r . ds) cos (r . ds') ] , 

 d z r 



or ^^ 



dsds 



since this vanishes when integrated round either circuit, the 

 two formulae give the same result when applied to entire 

 currents. A general formula including both that of Neumann 

 and that of Weber is evidently 



n'(ds .ds') .., ffr 



+ ki^ -j-, dsds, 

 r ds ds 



where k denotes an arbitrary constant.^ 



Helmholtz's result suggested to Clausius a new form for 

 the law of force between electrons ; namely, that which is 



* Journal fur Math., Ixxii (1870), p. 57 : Ixxv (1873), p. 35: Ixxviii (1874), 

 p. 273. t Cf. p. 229. 



+ Cf. H. Lamb, Proc. Lond. Math. Soc., xiv (1883), p. 301. 

 Journal fiir Math. Ixxxii (1877), p. 85 : Phil. Mag., x (1880), p. 255. 



