268 Prof. It. Clansius on the Determination of the 



a quantity which is the product of the just-determined mean 

 differential coefficient and the length of the piece of conductor 

 in question ; thus 



l/*'d.r'\ ds\ fi f 'bx'\ ds\ 



j'\ ds\ (if 1W\ 

 r 'd/J 1 dt T \r d*' A dt 



Consequently we have to put in the place of the preceding 

 •llowing sum — 



L JtlrW I -ds f ) 1 dt' 



integral the following sum 



But this sum is nothing else but the differential coefficient, 

 taken with respect to t, of the integral 



'i %' ~§x' , . 

 -^ T ds / , 

 r-ds' 



I 



if therein not only the quantity under the integral-symbols 

 but also the upper limit s\ be regarded as a function of t. 

 The alteration to be undertaken with the above integral con- 

 sists therefore only in this, that the differentiation there indi- 

 cated under the integral-symbol is to be indicated before it. 

 Moreover it must be remarked that the integral extended over 

 the whole of the closed circuit is not, like one referred to a 

 single conductor-element, to be looked on as a function of t 

 and /, but as a function of t only, and that hence, in indica- 

 ting the differentiation, d can be employed in this case instead 

 of §, so that the expression will be 



dt) n r~ds' 



ds f . 



»/'0 



Accordingly equation (9), when account" is taken of the 

 circumstance that the length of the conductor can change, 

 changes into the following, in which we will now, for simpli- 

 city, omit the limits of the integral (the adding of which was 

 expedient for the preceding consideration), because, after it 

 has once been said that all the integrals are to be extended 

 over the entire closed conductor, they are understood : — 



x^i a f£ 2 £ !£*•-*# f- It*'- . . (10) 



0%J r dt d» dij r d* 



In the same way, denoting by 3E X and £ 2 those values which 

 the same force-component must take according to Biemann's 

 and Weber's fundamental laws, we obtain from equations (6) 



