THE BALLISTIC GALVANOMETER 117 



and Q is given approximately by 



-) () " = ) "' ) 



Discussion. TF/ien /ie Discharge through the Instrument is 

 Prolonged. If the equation connecting e and t is known, the 

 integrations indicated in (5), (6) and (7) may be performed. 

 Suppose for example that 



e = 



In this case, which is of practical importance 4 , the total quan- 

 tity of electricity displaced in this circuit is 



Q 



- 1 (^dt = EQ 

 rjo rp 



Substituting the value of e in (5), 



_ f -' *' } CE 



2rPjb\p + mi p + m 2 j rP (p + w 2 )(p + r/ii) 



To find <i, the time of the first elongation or the turning point. 

 this value of 6 is differentiated and the result placed equal to 

 zero. Combining the equation so formed with (27) gives 



CE f 



01 = - - "Vi ~ "'. 



2rPj6p I 

 or after the values of mi and ra 2 are substituted, 



Equating -^ to zero gives 1, the time of the first elongation, as 

 a solution of the equation 



- p\T + 2X 2 + 2 



. 



The value of ti is obtained by successive approximation. 



Referring to (20) and (28) it will be seen that the ratio of the 

 deflection to that which would have been obtained had the same 

 quantity been discharged through the galvanometer instan- 

 taneously is 



