PHENOMENA OF INDUCTION. 607 



The velocity and the acceleration refer to the electrical 



Dr ^r 



masses, while the terms and - of the second member refer 

 of ot* 



to the distances of the two elements of the two circuits. 



The mechanical action of ds on ds' will be obtained as above 

 by taking the sum of the actions which the masses of the elements 

 ds exert on those of the elements ds'. With Weber's formula and 

 hypothesis, it is easily seen that in this sum there only remain, as 

 before, terms in z/z/, and with coefficients already found. It follows 

 from this, that in the variable state, the mechanical action is at each 

 instant conformable with that which Ampere's formula would give. 



625. The electromotive force which acts on the element ds' is 

 the force which tends to separate the equal masses of opposite signs 

 contained in this element, and to carry them in opposite directions. 

 We shall obtain the value by taking the difference of the actions 

 exerted in the direction of the element ds', on each of the masses 

 which it contains, by the two masses of the elements ds. But when 

 we add the actions of the two masses + m and - m of the element ds 

 on one of the masses m of the element ds t the terms which remain 



dv 



are terms in v, vv' and , which change sign at the same time as m. 

 dt 



Among these the only ones which remain in the final difference are 

 those which change sign with the velocity v, whatever may be the 



sign of v'. These terms reduce to two : one arising from ( J , 



. d^r *bu Dr 



and which is 2V = -^-, the other arising from , which is -r-r- 

 osot dt* ot Ds 



The difference thus calculated is equal to 



4mm' \~ DvDr Dr T)r~\ i F DIDr "tor Dr~\ 



r --- v -- = r --- I dsds' 

 a*r* L ^ fc fc *J r* |_ T>t Ds Ds *J 



taking into account equations (6), and supposing the intensity equal 

 to unity in the element ds'. 



We must take the component of this action along ds', and 



therefore multiply the preceding expression by ; observing that 

 we have 



lit <tf lit r 



