PEIRCE. — BALLISTIC GALVANOMETERS OF LONG PERIOD. 291 



elapsed and the angular velocity has become o>i, this velocity be im- 

 pulsively increased by the amount o> 2 , and a> are given during the 

 first stage of the motion by the equations 



= ( *. e - at -smpt, (5) 



P 



a) = e _a/ [w • cos pt sin pt], (6) 



P 



and 0! = — ■ e~^ ■ sin pt u (7) 



P 



Wl = e ati I w ■ cos ph sin p^ij. (8) 



p 



p = 2 ir/T, a = 2 A/7 7 , a/p = X/tt, /3 a = p 2 + a 2 . 



If, then, for 0' and </ in (3) and (4) we substitute 6 X and wi as given 

 by (7) and (8), and for t in (3) and (4) put (t — h), in order that the 

 origin of time shall be that of (5) and (6), we shall get 



6 = -° e-* • sin pt + - e~«W> sin p ft- fi), (9) 



P P 



w = w e~ a< [cos pt sin pt] 



P 



+ a) 2 e-««-'i) [cos p(t -h) --■ sin p - ti)]. (10) 



P 



Dorn points out that after the second impulse at t = fo, the motion 

 is the same as it would have been if there had been no such impulse, 

 but if when t — 0, the values of 6 and w had been 



— — • r'i • sin p*i, (11) 



and w + o> 2 • e ati [cos pti -\ sin pt{], (12) 



and shows that the formulas can easily be generalized to fit the case in 

 which there are a number of belated impulsive changes in the angular 

 velocity, instead of one. 



In the motion represented by (3) and (4), the angular velocity van- 

 ishes at the time t' defined by the equation 



