Transient Electromagnetically Induced Current. 279 



the time-integrals of the induced currents on the make and 

 break are equal is as follows "*. 



Let J3 and y be the currents at time t in the primary and 

 in the secondary circuits respectively ; J the self-inductance 

 of the secondary circuit ; and M the mutual inductance of the 

 two. We have 



Hence if y = when t = 0, and if we suppose R to be constant, 



J 



o y*-R0eo-/3)- E 7. 



Now let T be any value of t so large that /3 has become 

 sensibly constant, and 7 has subsided to zero. We have 



j: 



r M 



7*= 5 (/3 °' 



This shows that the time- integral of the induced current in 

 the secondary circuit would depend solely on the difference of 

 values of the current in the primary at the beginning and end 

 of the time included in the reckoning, and would be quite 



[* February 23. — Worked out more perfectly it shows, as follows, that 

 the common opinion is correct ! 



Imagine the whole secondary conductor divided into infinitely small 

 filaments of cross-section d€l : and let £ he the current-density, at time t, 

 in any one of these. Instead of y in the text take gdQ, and instead of R 

 take ll(o-dQ.) ; o- denoting the specific conductivity of the material, and I 

 the length of the circuit. We have 



at at 



where S denotes the sum of effects due to the risings and fallings of cur- 

 rent in the different parts of the secondary conductor. Their time-integral 

 from to T is essentially zero ; as is also that of the infinitesimal middle 

 term of the second memher. Hence we have 



/o--ij o T ^=M(/3-/3 ); 



which shows that the time-integral of the current, £dQ, in each JUament 

 of the secondary conductor, is exactly equal to that which is calculated 

 according to the ordinary elementary theory ! The whole details of the 

 fallacy in the text are now clear !] 



