Short-Period Electrometer. 251 



(9) Influence of the Brass Tube on the Period of Oscillation, 



. From the dimensions of the brass tube on which B was 

 wound and the mutual inductance of A and B, the self- 

 inductance of the tube and the mutual inductance of it and 

 A can be approximate^ calculated. 



Calling these L 3 and M 13 respectively, the factor 

 1 — M^/LxLg was found to be '0229. It might appear 

 therefore that the presence of the tube would considerably 

 reduce the effective self-inductance of the coil A and diminish 

 the period of oscillation by rather more than 1 per cent. 

 The resistance of the tube is. however, very great in com- 

 parison with its self-inductance and must be taken into 

 account. 



Calling the integral currents in A and the tube x and z 

 respectively, the equations for these two circuits are 



Eliminating z .and assuming #=g (a+ *^*, where /3 = 27r/T, 

 and a. is the damping factor, we obtain the equation 



(LiLs - M 13 2 )(<* + *'/3) 3 + (L.R, + LA) (« + *73) 2 



+ (^+R 1 E 3 )(a+i/3)+ R 3=0 > ... (4) 



for the case in which the coil B is open. 



The specific resistance of brass being assumed, to be 5000 

 c.G.s., the numerical values of the coefficients were inserted 

 in the equation which was then solved for a-\-i/3 by Cardan's 

 formula. There is no difficulty in determining which of the 

 three roots suits the case, and thus finding a and ft. 



For the case of the largest condenser (IV.) the results 



were 2tt 



a= -104-5, T= — =-003761 sec. 



Equation (1) gives in this case T = -003762 sec, while 



^- has the value 99'8. 



The brass tube has therefore very little effect on the period 

 under these circumstances. 



With Condenser I. the cubic gives 



a= -117-6, T = ^ =-001711 sec, 

 P 



while equation (1) gives T = '001718 sec. The tube appears 



S 2 



