430 Prof. A. Gray : Notes on 



Multiplying (4) by 7 and subtracting from (5) we get 



mvv =-^-Ly 2 ; (6) 



so that the time-rate of increase of the kinetic energy of the 

 slider is equal to the rate of increase of the elect rokinetic 

 energy. 



This is of course in accordance with the theorem given by 

 Lord Kelvin in 1851 (Electr. and Mag. 2nd ed. p. 446) 

 with regard to the work done in two mutually influencing 

 circuits, the conductors of which are brought to rest by 

 externally applied forces. 



It may be noticed that (6) may be obtained at once from 

 the expression for the whole kinetic energy 



^Ly 2 +\m'x 2 



(where x = v) by the Lagrangian method. For we have 

 then 



or 



mxx— !Ly 2 = 0, 

 which is (6). 



Now the self-inductance of the rails regarded as direct 

 and return wires of circular section connected by the slider 

 at one end can easily be found. If x be the length of rail 

 from the slider to the points of attachment of the wire, p the 

 radius of each wire, I the distance apart of the axes of the 

 wires, and the magnetic inductivity be taken as unity every- 

 where (wires made of copper, for example), the self-inductance 

 is given by 



L = 4i' ( log - -J- j I — 4/ -j- terms depending on the rail cross- 

 \ P connexions. 



Hence we have 



or 



d^ A ^ I . 1 

 — =41og- +1, 

 dx & p 



dh dL . 



A=^ = ( 41og p +1 r • • (/) 



Now we have seen that onvv = ^Jjy 2 , so that from (6) and 

 (7) we obtain 



mvv=-iU]og- + l\ 2 v, .... (8) 



