184 SOURCES OF ELECTRICITY. 



machine and on its position in reference to the adjacent conductors ; 

 in the former case, it depends on atmospheric conditions and on the 

 rapidity with which the operations succeed each other. 



With addition machines, the limit is attained when, at any time, 

 the gain is equal to the losses, and this limit would always have a 

 finite value. 



In the case of multiplication machines, certain conditions must 

 be fulfilled if the charge of the receiver is to retain a finite value. 

 Let C and C' be the capacities of the two receivers, V and V 

 the potentials in absolute values, c and c' the capacities of the 

 carriers, and n and ri the number of operations performed in unit 

 time. 



The loss by air of an electrified conductor is sensibly propor- 

 tional to the charge, or to the potential, for very feeble charges. 



If m and m are the coefficients of the proportionality relative to 

 the two conductors, the losses of the charge for unit time might be 

 represented by mV and m'V. 



During an infinitely small time dt, the increase of charge of the 

 receiver C is equal to the excess of the electricity which it receives, 

 over that which it loses ; we shall have then 



or 



C = V'V'-wV. 

 at 



The other receiver will give, in like manner, 







at 



Solving these simultaneous differential equations, we could calculate 

 the values reached by the potentials V and V at the end of a given 

 time, starting from given initial values ; but the results thus obtained 

 would only hold within the limits in which Coulomb's law may be 

 admitted. We know that for somewhat larger charges the loss takes 

 place according to a more rapid law. 



These equations give the conditions necessary for the charge to 

 go on increasing ; for this it is sufficient if the differentials of the 

 potential are positive, which gives 



n'SV'-mVX), 



