PEIRCE. — BALLISTIC GALVANOMETERS OF LONG PERIOD. 297 



After the time t — t, Mt, Nt have the constant values 31, N, and 

 Qi becomes Q, the total amount of electricity carried by the current 

 from t = until it ceases to flow at t = r, so that 



= -$- [M- e mt - N- e nt l (29) 



m — n 



If, as is usually true in practice, ft is greater than a, p is posi- 

 tive, m = — a + pi, n = — a — pi, m/(m — 11) = ^ + ai/2p, 

 n/(n — m) =% — ai/2 p, but the results are, of course, real. 

 If we determine dO/dt from (29) and equate it to zero, we learn that 

 at a time of elongation 



t — - 



m 



and this value of t substituted in (29) gives the amplitude at elonga- 

 tion in the form 



A = I ( Yin— n I \m— n \]\fn — m . ffm—n (31) 



in — n \_\m J \m J v ' 



m 



= C ■ M n - Jm ■ N 11 ™ (32) 



where C is a function of the constants of the galvanometer and is inde- 

 pendent of the manner in which the whole flux Q of electricity is sent 

 through the circuit. If ^1 denote the amplitude at the first elonga- 

 tion when Q is sent impulsively through the coil at the origin of time, 



m 



A JL - 



— = M n ~" 1 ■ N m -^' (33) 



If / happens to be given in analytic form as a function of t, it is possible, 

 as Diesselhorst shows in a general case, to obtain a convergent series 

 for A/A . For the purposes of this paper, however, where the form of 

 / is shown merely by an oscillograph record, we shall find it desirable 

 if m and n are real, to plot the curves y = Ie~ mt , y = Ie^ 1 ' directly 

 from this record and then to find the values of iWand N by mechanical 

 integration. 



j &-* 



If ft is greater than a, (27) may be written 

 = - e~ ai [sin pt- I I e at ■ cos pt-dt — cos pt- I I e at ■ sin pt • dt\ (34) 



p Jo .70 



