FORMULAE OF GAUSS AND WEBER. 605 



In order to satisfy Ampere's law, it is necessary that, in the first 

 case, 



and, in the second case, 



a*p = r, 0*0.'=--. 



2 



The expressions (i) and (2), which give the elementary action of 

 the two electrical masses, become then 



dt 



i/ d*r i /dr 

 I r 



\~\ Ti 2 d**Jr~\ 



I =mm\ I- - 



/ J L r2 Vr dt<L 



622. The former expression (i)', which occurs in Gauss's 

 manuscripts, is incompatible with the principle of the conservation of 

 energy, for it would lead to the conclusion that a limited physical 

 system can produce an indefinitely increasing quantity of energy. 



Weber's formula, on the contrary, is compatible with this prin- 

 ciple ; for the expression of the force (2)' may be considered as 

 the differential coefficient in respect of r, taken with the contrary 

 sign, of the function 



(7) </>= - 



The work done by the repulsion of a fixed mass on a movable 

 mass is equal to the difference \l/ - ^ of the values of the function ^ 

 relative to these two masses, for the initial and the final position. 

 The function ^ may be considered as representing the potential 

 energy of the system of the two masses ; it only depends on their 

 distance, and on their relative velocity along the right line r; it 

 resumes the same value when one of the masses describes a closed 

 path in reference to the other, and possesses the same velocity at the 

 same points. 



Since induction is a consequence of the law of Ampere, and of 

 the principle of the conservation of energy, Weber's formula, which 

 equally well satisfies both conditions, must give the induction. 



