PEIRCE. — BALLISTIC GALVANOMETERS OF LONG PERIOD. 311 



and these expressions involve A and 8 but are independent of the 

 sensitiveness of the galvanometer and of its time of swing. 



It is possible to show from equations (44) and (45), after some com- 

 putation, that for the case of the galvanometer V, for which we may 

 take a = 0.0125, p = 0.0450; A"/A = 0.994, or 0.998, according as r 

 is 8 seconds or 4 seconds. It is well to recall the fact mentioned above, 

 that A' I A = 0.982 or 0.995, according as t = 8 seconds or 4 seconds. 



Perhaps most of the induction currents which one meets in making 

 magnetic measurements have forms similar to those of the curves S or P 

 in Figure G, and it is worth while to compute the value of the ratio 

 A/A on the supposition that the current flows from t = to t = t 

 with the intensity I—k ( t — t) where it is clear that k — 2Q/t 2 . 



Since 



/e Kx 

 x-e^-sinx-dx = /, , A 2\a C( A ' sm x *~ cos X )Q^ X + x — A ) 



+ (sin x + A • cos x)], (48) 

 and 



x-e**- cos x-dx = . 2 2 [(sin x + A • cos x)(\ 2 x + x — A) 



— (A • sin x — cos x)\ (49) 



it is not difficult to prove that when I = k(r — t), 



2 

 R = 754 — r, [a • e aT (p • sin pr + a • COS /dt) 



+ p ■ e aT (a • sin pr — p- COS pr) + p 2 — a 2 — a^3 2 T], (50) 



2 

 S = 754 — j [a • e aT (a • sin pT — p • COS pr) 



— p • e aT (p • sin pr + a • COS pi") + /3 2 pr + 2 ap]. (51) 



These formulas are not very well adapted for easy computation, and 

 in many practical cases in which the quantities in the brackets are 

 very small and the coefficient 2//3 4 t 2 very large it is desirable to use 

 five or six place logarithms in the work. As an illustration of the 

 use of these equations we may consider the instance of the galva- 

 nometer V through which a current of the form 1= k (t — t) shall 

 flow for 8 seconds. Here a = 0.0125, p = 0.0450, /3 2 = 0.0021812, 

 2//3V = 6568.39, R = 1.04723, S = 0.12545, and A/A = 0.9974. 

 The throw due to this current is the same within about one quarter of 

 one per cent as if the whole amount of electricity conveyed by the cur- 



