802 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Theory of Action of Crystal as a Leaky Condenser on the 

 Assumption that Charge and Discharge are Complete. 



R 



I— VW\AA 



Let us suppose an e. m. f. E (Figure 9) to be applied to a condenser 



C through a resistance R, and suppose that 

 the condenser is shunted by a resistance r, 

 through which the leak occurs — the resis- 

 tances being of such a character that the 

 current through it is not proportional to 

 the voltage. 



Let a- = the feed current in the main 



lead R, 



y = the leak current through r, 



V = difference of potential between the 



ends of the resistance r ; this is 



a function of y, and is also the 



FiGUBE 9. 



p. d. between the plates of the condenser. 



These quantities, .r, y, and v are functions of the time t. 

 Kirchhofifs laws give the following equations : 



E=Rx-\-v, 

 obtained by taking the e. m. f. around the circuit ERCr. ; and 



(x — y)dt = Cv, 



/< 



(1) 



(2) 



since [x — y) is the current charging the crystal capacity. 

 By integration of equation (2) from ^ = to ^ = T, we have 



xdt = C{vt — Vo) + / ydt, 

 Jo 



(3) 



in which qi is the quantity of electricity flowing through the charge 

 leads during the time T of one complete connection of the condenser 

 to the charging e. m. f. 



If we suppose the previous discharge of the condenser to have been 

 complete 



vq = 0. (4) 



If also we suppose the charge of the condenser is complete in the time 

 T, the current x flowing through the leads at time T is Y, where Y is 

 the steady current through the crystal under the impressed e. m. f. E ; 

 therefore, by equation (1), we have 



= E-RY. 



(5) 



