298 PROCEEDINGS OF THE AMERICAN ACADEMY, 



If R ■ Q = C T J. e «t . cos P t ■ dt and 8 ■ Q = Cl- e ai • sin pt > dt, (35) 

 the value of 6 after the current has ceased is 



S ^^QL-lfi.sinpt- 8 C os pt] (36) 



r 



where Q, R, and 8 are constants. 



At the first elongation, 



, pR + a-8 . > 



tan '' = ^^^ ( 37 > 



a.R — p8 . pR + a.8 . x 



or cos p£ =. — .,- sin p£ = , : , (38) 



PVR* + S* (3VR 2 + 8 2 



and if the first root of these equations be substituted for t in (36), it 

 appears that the first elongation is given by the expression 



i^^ (39) 



where u = - • tan -1 "—^ - y . (40) 



p all — pb 



If the quantity Q of electricity had been sent impulsively through 

 the galvanometer when the coil was at rest in the position of equi- 

 librium, the throw would have been as (5) shows 



A = f-e-« (41) 



where v = - • tan -1 - . 



p a 



A 



Hence ~ = V R 2 + 8 2 ■ e"<«-*l = VR 2 + & 2 • e^>, (42) 



where w — -tair'-T-,. 

 P R 



If \ Q were sen t impulsively through the circuit at t = 0, and £ ^f 

 at £ = t, the values of R and # to be used in (42) would be 



R = i (1 + C aT ' COS pr), £ b £ e OT • sin pr. (43) 



