Ponder omotive and Electromotive Forces. 267 



takes place, causing parts of the conductor which were pre- 

 viously outside of the circuit to be afterwards within it, or vice 

 versa. In the parts which in this process are added the cur- 

 rent commences ; in those withdrawn it ceases ; and by this 

 alteration of the current-intensity in particular parts of the 

 conductor a force is conditioned, which must also be taken 

 into account. It is true that, on account of the great velocity 

 with which the commencing and cessation of the current are 

 accomplished, the parts of the conductor in which at any mo- 

 ment they take place are very small; but the differential coeffi- 

 cient — for them is very great, and through this the corre- 

 sponding portion of the force may take a considerable value. 

 The question now is, how this part of the force can be also 

 expressed in the formula. 



We will choose the places where the entry and exit of parts 

 of the conductor take place as the initial and final points 

 respectively of the closed conductor /, so that a newly enter- 

 ing piece of conductor is annexed exactly at the end of the 

 conductor. If s\ denote the length of the conductor at the 

 time t } the element of conductor added during the time-element 



will be represented by —Adt. If, further, the very short time 



requisite for the production of the current in a conductor-piece 

 entering the closed circuit be denoted by t, then will, during 

 the lengthening of the conductor, a piece at the end of it of 



ds f 

 the length — ^ r be that in which the origination of the cur- 

 rent takes place. This origination is an increase, taking place 

 during the time t, from to the value i r prevailing for the 

 rest of the conduction. The mean value of the differential 



coefficient ^- in this piece during the time r is consequently 



i' ^ 



= — ; and so we can represent the corresponding mean value 



of the differential coefficient ^— ( — ^— r) by -( ~^ )> wherein 



the index 1 put to the brackets is to indicate that the quanti- 

 fy 

 ties r and ^-7- within the brackets have the values belonging 

 to s\. ^> s ' .... . 



Now, in order to bring likewise into calculation in our for- 

 mula the origination of the current in this small piece of con- 

 ductor, we have to add to the second integral in the formula, 

 which if we write it with the limits has the form 



U2 



