PEIRCE. — BALLISTIC GALVANOMETERS OF LONG PERIOD. 299 



With some of the forms of short period, critically damped d'Arson- 

 val galvanometers commonly used in American laboratories, it is diffi- 

 cult to reverse the current in the primary of an induction apparatus 

 with air core by a large double throw switch so quickly as to avoid a 

 decrease in the throw of the galvanometer coil owing to the lag in the 

 second impulse. 



If a current of constant intensity (Q/r) flowing for the time inter- 

 val t conveys a quantity, Q, of electricity through the circuit, the 

 values of R and 8 are 



R = o2" [e aT (p • sin pr + a • COS pr) — a] (44) 



S = -™- [e aT (a sin pr — p • cos pr) -f p] (45) 



^/R 2 + S 2 = — Ve 2aT - 2 e aT cos pr + l. (46) 



In the case of a critically damped instrument 



6 = fxe~ [t I I-e at -dt- lit- e at -dt\. 



If there were no damping, a would be zero, e~~ w would be equal to 

 unity, and R and S would satisfy the equations 



RQ= I I ■ cos pt-dt, SQ= I I ■ sin pt-dt. 



Jo Jo 



The foregoing theory rests, of course, upon the assumption that the 

 swinging system of a galvanometer meets with a resistance to its mo- 

 tion which may be attributed to a force couple of moment equal at any 

 instant to the product of a fixed constant and the angular velocity 

 which the system then has. It is evident, however, that this condition 

 cannot be exactly fulfilled during the whole motion of the needle or 

 coil of any instrument in which the damping soon brings the swing- 

 ing system absolutely to rest. In the case of a horizontal bar magnet 

 swinging without sensible friction about a vertical axis through its 

 centre, the ratio of successive half amplitudes usually remains nearly 

 constant for a large portion of the motion, though the actual value of 

 the ratio often depends upon the atmospheric conditions, as Gauss 

 showed. The logarithmic decrement of the oscillations of a magnetic 



