536 



SCIENCE. 



[N. S. Vol. I. No. 20. 



The e. m. f. should therefore be introduced 

 and removed without breaking the circuit. 

 If we assume no damping action on the 

 needle the equation to its motion is 



where n is a constant depending on the gal- 

 vanometer and the intensity of the mag- 

 netic field at the needle, while X depends 

 on the galvanometer and on the nature of 

 the transient current. If we suppose the 

 impulse given to the needle to be due to 

 the charge or discharge of a magnetic field 

 and take the permeability of the core as con- 

 stant we may put X = A e« ^ where A is 

 a constant depending on the galvanometer 

 and « = L where li is the resistance and L 

 the co-ef&cient of induction. 



We thus get ^ 



df 



+ nW = A e-« f 



The solution of this equation is 



S = -=— ; ; \ e~° '-| — smnf-oosnf [ 



where term 3 ^- ■ 



' a 



The constant n is equal to 2,t/T, where T 

 is the free period of the needle. 



Take, as a particular case, a ring of mean 

 circumference Z = 30 centimetres, and cross 

 sectional area S ^ 2 square centimetres, and 

 suppose the total number of turns on the 

 magnetizing coil to be N" = 600, the per- 

 meability /i = 2000, and the resistance 1 

 ohm. Then the increase or decrease of in- 

 duction per unit current X IS" ^ L ^ 



1 y 109 = A nearly in henrys. Hence 

 we have a or E/L ^ Jg"-, and the current at 

 time t, after the removal of the e. m. f., the 

 circuit remaining closed, is Ct = C„ e-V t 

 where Co is the current just before the 

 e. m. f. is removed. Giving t different val- 



ues in seconds we have the following values 

 of the ratio Ct / Co : 



t in seconds =1 2 .3 4 5 ~ 

 Ct / Co = 0.1889 0.03565 0.00673 0.00127 0.00024 



If the resistance be taken equal to 10 ohms 

 then the unit of time in the above table is 

 to be taken as one tenth of a second, and so 

 on for different resistances. Preciselj' the 

 same calculation appUes to the case of in- 

 creasing magnetization, only Co is then the 

 fijial steady current, and the numbers in the 

 line Ct / Co are the differences from unity 

 of the ratio Ct / C^^ , that is, the equation 

 becomes Ct / Co = 1 — e— b- t. 



Hence, remembering the high value which 

 L may have at certain parts of the cycle in 

 the case of iron, we see that to insure the 

 whole quantitj^ of electricity getting through 

 the galvanometer coil in a small fi-action of 

 the quarter period the resistance would re- 

 quire to be in the neighborhood of 1000 

 ohms for a needle of 4 seconds period, and 

 of 100 ohms for a needle of 40 seconds 

 period. 



The quantity of electricity which flows 

 through the coil in time t is given by the 

 equation 



-/"• 



(--i.) 



Hence in the case supposed above the 

 quantity which flows in one second is about 

 I of the whole when the resistance is one 

 ohm, and about -|- of the whole in j^jj of a 

 second when the resistance is 100 ohms. 



The equatim e ^^J e'^ ' + ^j'!!_+^ sin 

 («'—/?) h reduces to 6' = an in the case 



of a being very great in comparison with n 

 and this form can be readilj' reduced to the 

 equation commonly given on the supposi- 

 tion of tlie time of discharge being small in 

 comparison with the period of the needle. 

 Keeping to the case taken above of the 



