﻿and other kinds of Ballistic Galvanometers. 271 



Moving-needle instruments will now be considered. The 

 formulas 



n HT sin 0/2 . n T* sin 0/2 



Q = — p — and Q= — — ir L - 



apply to what may be called a tangent ballistic galvanometer. 

 The time of the transient current must of course be small 

 compared with T, and its greatest value mast not be sufficient 

 to appreciably change the magnetism of the needle. 



In many ballistic galvanometers an astatic system of nee lies 

 is used. First, suppose the system is exactly astatic and that 

 the torsion of a wire or quartz fibre supplies the controlling 

 couple. In this case. Ci cos <p = xcf) and Kor = a#-. so that the 

 formulae are 



O _T«0 _ Tiflc os(ft 

 ^""2ttC 2tt£ ' 



as with the second type of moving-coil instrument. Also 

 C=M(G l + G 2 ), where M is the moment of each needle and 

 Gi and G 2 the fields due to unit current at the two needles 

 respectively. 



One other case will be considered, viz. when the needle 

 system is more or less astatic and is controlled by magnetic 

 fields at each needle. Let the horizontal component of the 

 controlling field at one needle be H x making an angle ot. l with 

 the plane of this needle, and let Mj be the moment of this needle. 

 Let M 2 , H 2 , and a 2 be the corresponding quantities at the 

 other needle. Then in the equilibrium position, 



= MjHi sin «! + M 2 H 2 sin « 2 . 

 The controlling couple when the needle is deflected through 

 an angle <f> is 



MJi x sin (</> + «i) + M 2 H 2 sin (<p + « 2 ) 



= {MiHj cos a x + M 2 Ho cos a 2 } x sin <fi ; 

 and if <£ is the deflexion due to a current i, then 

 iMxGi cos </> + iM 2 G 2 cos <£ = sin ^{MiHi. cos a x + M 2 H a cos u 2 }. 

 Hence 



MjHi cos «! + MoHo cos ^ 



i= m,g 1+ m:gI _tan *- 



It can now be easily seen that the formulae for a ballistic 



galvanometer of this type are 



n _ TCM^cos^ + MaH oCos « 2 ) sin 0/2 _ Ti sin 6/2 



^ " ~ ^(M^ + M 2 G 2 ) ~ and ^ ~ 7rtan^> * 



So far the correction of the observed deflexion for 



damping has been neglected. The method usually described 



depends on the " logarithmic decrement," and is cumbrous, 



and moreover inexact unless the damping is very small. The 



