[CLARK] CAPILLARY ELECTROMETER 79 



Ciq2 , ^ T3 dq2 



or qi =-— L + CiRa-^ 



C2 at 



We may differentiate this equation and substitute in (1), obtaining 



CiR,R2 + L + t^^ R, + R, + R,^ (3) 



d^q2_. ^^_^_| C-2dq2 , q2 ^q 



dt2 L Ci R2 dt2 L Ci R2 dt LC1C2R2 



We may apply the general method for linear differential equations 

 with constant co-efficients by placing q2 = €'^^, substitution of which 

 in (3) gives 



CiRiR2 + L + L^^ R, + R2 + R,^ <^^) 



P + -'X' + 2!A + —J = 



L Ci R2 L Ci R2 LC1C2R2 



yielding three values of A, so that 



q2 = aie^it + ^2 €^^^ + aje^^t (5) 



It will be shown later that only one root of the A equation is real. 

 The other two must be conjugate imaginaries. We may call Xi = A, 

 the real root, and the other two may be written a -\- 'i/S and a — i/3. 

 So we have finally 



q2 = ae^t ^ ^«t (^ cos |9t -|- B sin (3t) (6) 



Differentiation and substitution in (2) give 



<, = a(§-; + C.R,A)e^t ^^^ 



+ e«t |piA+CiR2(Aa+B/?)|cosi3t+|p^B+CiR2(Ba-A/?)|sin/3tj 



We may, of course, write (6) as 



q2 = ae^t 4. Ae«t cos (/?t - 0) (8) 



where A = ^ A^ + B^ and = tan"' - (9) 



and in the same manner 



qi = ( ^ + CiR2A)ae'^t+Aie«tcos(/?t-0 + w) (10) 



where Ai = A ^ (^ -' + Cj R2 aV + Ci^ Ra^ (S' 



and ^ = tan-^ ^'^'^ (11) 



1 - C2 R2 a 



(Ù is the difference in phase of the two discharges. It will be noticed 

 that both discharges are oscillatory and consist in each case of a 

 damped oscillation impressed upon an exponential discharge. 



