812 PROCEEDINGS OF THE AMERICAN ACADEMY. 



charge and discharge leads to the result that the capacity is indepen- 

 dent of n but is dependent, as in Experiment I., upon the magnitude 

 and direction of the impressed e. m. f. We shall give a brief deriva- 

 tion of the formula for multiple incomplete charge and discharge. 



Sketch of Theory of Multiple Incomplete Condenser 



Discharge. 



Let a condenser of capacity C be alternately charged and discharged 

 through a resistance R which is so great as to permit only a partial 

 charge or discharge in the time T of one connection of the condenser 

 into the charge or discharge circuit. 



Let the applied e. m. f. on charge be E, and suppose that the charge 

 and discharge have been repeated until a final state is reached. 



Let qi = quantity of electricity in the condenser at the end of a charge, 

 qo = the quantity in the condenser at the end of a discharge. 



If we write down the differential equation for the quantity in the 

 condenser on charge and integrate it subject to the conditions of the 



problem, we obtain 



r T 



q, = goe"^^ +CE{\- e"«^). (24) 



Similarly from the differential equation for discharge we get 



T 



go = qii''''. (25) 



Eliminating between equations (24) and (25) we have 



■ CE 



qi = -^^' 



1 + e'""" 



Qo 



T 



CEe''"'' 



1 + e 



T 

 ' RC 



The quantity of electricity flowing into the condenser during one 

 charge, or out of the condenser during one discharge, is 



T 

 ' RC 



q^q^-q,= CE- ^. (26) 



1 -I- e"''^ 



