128 Dr. 0. J. Lodge on Intermittent Currents 



obtained by the solution of a linear differential equation of the 

 third order with constant coefficients. 



But the problem may be simplified by observing that the 

 " coin " is not introduced until a perfect "balance " has been 

 obtained between the first and second circuits — that is, until 

 they are conjugate to one another so that M 12 = 0. Moreover 

 the coefficients of induction between the two coiled circuits 

 and the coin are ordinarily sufficiently small for their squares 

 to be neglected ; in which case, as will be shown more fully 

 later (sections 12 and 13), the above set of equations reduces 

 to the following — 



m u §+ o + o +fA=m, 



+M 22 | 2 + M 23 J+r 2 i 2 =0, 



M 4 1+ 0+M S3 f + ^=0, 



or, as I shall for greater convenience write them, 



,dj , dk , . . . . . . . (2) 



l dt +m dt +r 3 = °> 

 dk di -' A 



x di +ll dt + P k=0 >) 



where the three coefficients of self-induction are written L, I, 

 and \ (X being very small); and where the battery-current is i 

 the telephone-current is j and is small, and the current in the 

 coin is k ; m signifies the effect of the coin-current on the 

 telephone-circuit, and (ju the effect of the primary circuit on 

 the coin. The effect of the primary on the secondary (M) is, as 

 already explained, supposed to be ; that is, balance is supposed 

 to have been obtained before the insertion of the coin, p, or 

 the resistance which the current circulating in the coin experi- 

 ences, is a quantity which is likely to be several times larger 

 than the specific resistance of the material (i. e. the resistance 

 of a centimetre cube) ; but for ordinary coins it is a number 

 of the same order of magnitude. It must be remembered that 

 R and L stand for the resistance and self-induction of the 

 whole of the battery-circuit, and, similarly, r and I include the 

 telephone-coil as well as the other two coils of the telephone- 

 circuit. 



Now the only difficulty in solving these equations consists in 

 the varying resistance of the battery-circuit, in which a con- 



