290 PROCEEDINGS OF THE AMERICAN ACADEMY. 



1301 X 10~ 6 in 422 oscillations. It will appear in the sequel that the 

 two long period galvanometers described in this paper follow the 

 Gaussian law, if not exactly, still quite nearly enough to make it worth 

 while to study their characteristics in the light of the usual theory. 



The behavior of a damped ballistic galvanometer through which im- 

 pulsive currents flow when the suspended system is away from its posi- 

 tion of equilibrium and is in motion was first treated thoroughly by 

 Dorn in a paper 7 written before d'Arsonval galvanometers were much 

 used. In this paper Dorn studies- the error introduced into observa- 

 tions made by Weber's methods of multiplication and of recoil, when the 

 impulses are not properly timed. He also considers the case where the 

 galvanometer is subjected to the action, not of a series of impulses, but 

 of a continuous current, which lasts with given varying strength for a 

 considerable time, and some of his equations have lately been put into 

 other convenient forms by Diesselhorst. We snail find it desirable to 

 derive from the beginning the special equations which we need in this 

 paper. 



The equation of motion of the coil of a d'Arsonval galvanometer, 

 when the resisting moment is proportional to the angular velocity, is of 

 the form 



where K is the moment of inertia of the suspended system about the 

 axis of suspension. If this equation be written in the form 



a may be called the " damping coefficient," and /? 2 the " restoring coef- 

 ficient." It will be convenient to represent dd/dt by w, (/3 2 — a' 2 ) by p 2 , 

 and the complete time of swing of the coil by T. 



If when t = 0, 6 and w have the given values & and <*/, the general 

 solution of (2) takes the form 



6 = tr* \& • cos pt + W ' + a6 ' ■ sin P t\ (3) 



P 



i t r i aoi "+" fi v' . , 



whence w = e~ at [u ■ cos pt sin pt}. (4) 



r 



If, when the system is at rest in its position of equilibrium, an im- 

 pulsive angular velocity w be given to it, and if after ti seconds have 



' Dorn, Ann. der Physik, 17 (1882) ; Diesselhorst, Ann. der Physik, 9 (1902). 



