on Galvanometers. 69 



logV=-ilog(l+^ 2 ) + ^ t an- 1 ^, 

 where x — — . 



TT 



Differentiate both sides of this equation once, twice, and thrice 

 with respect to x and substitute for x the value 0. We then 

 obtain successively, 



dV_n ^!Z_^_o d 3 Y _ 1 3 9 



das " 2 ; d.* 2 ~ 4 ; ^ ~ 8 W 2 """ 



By Maclaurin's theorem we then have 



it 4 TT 



♦(?-#+G--!-)?**« 



or substituting for #, and working out numerically, we 

 obtain 



V=l + 0-5\-(H)27A. 2 -005-4\ 3 , 



which shows that until X approaches unity the approximate 

 value for V generally used is sufficiently accurate. Now if 

 the ratio of the amplitude of any swing to that of its successor 

 is (1+y) we have 



\=log(l + t,)=y-|V|- 3 -&c., 



Neglecting powers higher than the third, we obtain by sub- 

 stitution 



V=l + 0%-O277 2/ 2 + O130?/ 3 , 



from which we can at once calculate the value of the complete 

 correcting factor when we have observed the decrement. 



It is usual to determine the sensibility of a ballistic galva- 

 nometer either by the employment of an earth-inductor, or 

 by discharging through the galvanometer a condenser charged 

 to a known P.D. The first method, however, necessitates an 

 exact knowledge of the horizontal or vertical component of 

 the magnetic intensity at the spot, while the second requires 

 an exact knowledge of the value of the capacity of the con- 

 denser and of the P.D. employed. But now that it is possible 

 to obtain an ammeter calibrated with a high degree of 

 accuracy, the simplest method of determining the sensibility of 

 a ballistic galvanometer is to first calibrate it absolutely as a 



