262 The Mathematical Electricians of the 



obtained by supposing that two electrons of charges e, e', and 

 velocities v, v', possess electrokinetic energy of amount 



ee f (v .v') 7 , d~r , 



- - + kee - r =-> w . 



r dsds 



Subtracting from this the mutual electrostatic potential energy, 

 which is ee'c'/r, we may write the mutual kinetic potential of 

 the two electrons in the form 



(xx + ijy + zz f - c 2 ) + kee' > vv', 



where (x, y, z) denote the coordinates of e, and (X, y', z) 

 those of e f . 



The unknown constant k has clearly no influence so long as 

 closed circuits only are considered: if k be replaced by zero, 

 the expression for the kinetic potential becomes 



ee' 



(xx + yy + zz - c 2 ), 



which, as will appear later, closely resembles the corresponding 

 expression in the modern theory of electrons. 



Clausius' formula has the great advantage over Weber's, that 

 it does not compel us to assume equal and opposite velocities 

 for the vitreous and resinous charges in an electric current; 

 on the other hand, Clausius' expression involves the absolute 

 velocities of the electrons, while Weber's depends only on their 

 relative motion; and therefore Clausius' theory requires the 

 assumption of a fixed aether in space, to which the velocities 

 v and V may be referred. 



When the behaviour of finite electrical systems is predicted 

 from the formulae of Weber, Eiemann, and Clausius, the three 

 laws do not always lead to concordant results. For instance, if 

 a circular current be rotated with constant angular velocity 

 round its axis, according to Weber's law there would be a 

 development of free electricity on a stationary conductor in the 

 neighbourhood ; whereas, according to Clausius' formula there 

 would be no induction on a stationary body, but electrification 



