606 GENERAL THEORIES. 



623. This formula also enables us to obtain directly the relative 



potential energy of two closed circuits. For, if we replace by its 



at 



value as a function of the velocities of the electrical masses, we get, 

 from Weber's hypothesis, that the potential of one of the elements 

 on the other is expressed by 



, ds') = - ITdsds - . 

 r 



The potential energy of the two circuits is then 



COS 



dsds . 



This is the formula of Neumann, as found above (353). 



When the currents are constant, and traverse circuits of constant 

 form, the resultant of the actions exerted by one of the currents on 

 any mass m' whatever of the other is perpendicular to its trajectory. 



624. PHENOMENA OF INDUCTION. Let us now suppose that 

 the currents move, and that the strengths change. The distance r^ 

 instead of being, as above, simply a function of two independent 

 variables s and s', is further a function of the time, and we have 

 r=<f>(s,s',t). For a given value of /, the function <j> represents the 

 distance of the two elements of the circuit ; if / is variable, and we 

 consider s and s' as functions of /, the value of <j> represents the 

 distance of two electrical masses in motion on movable conductors. 

 If the conductor has a constant section, the velocities do not at all 

 depend on s and s' as independent variables, for at each instant the 

 strength is the same in all parts of the circuit. Hence, for the 

 relative velocity of the two masses, we shall have 



dr ^r ^>r ^r 



and, considering the differential coefficients and as functions 



^s V 



of s and s' alone, the relative acceleration will be 



d*r JPr ,TPr , dV ^v^r V> Tnlr Wbr *&r 

 _ = , 2 _ +2Z;z; _ + z;2 __ 2 + __ + ___ + ,__ + ^___ + __. 



