and the Theoi*y of the Induction-balance. 135 



di 

 (18) so as to eliminate — 3 and then put in the values of j and 



^ from (21). We thus get 

 at 



M(Sz~E)=iJ^ a ^v / (^--S0 2 + 4rSM 2 .cosh^ 



+ (Lr-ZS)sin]i#}, 



which will give us the value of J, since i= ^ when £=0. 

 We find then that K 



2ME(S-R) 



°--R x /{(Lr-l8) 2 + 4,rSM: 2 y ' ' ' ' {Z } 



and this is the value to be substituted in equation (21). 



Hence, finally, we can write out explicitly the values of i 

 and j, though, as the constants are very long, we will make 

 an abbreviation by writing 



4rS %=*; (25) 



(Lr-i&y 



then the strength of the battery- current at any time after the 

 resistance of its circuit has suddenly jumped from R to S is 



and the strength of the induced current at the same instant 

 is 



^igl^r^- • • (27) 



We may notice that the expression for i contains only 

 the square of M ; that is, the square of M expresses the re- 

 action of the secondary on the primary ; hence when M is 

 Nsmall this may be neglected. 



Special Cases. 



' 13. It will be interesting now to consider the special cases 

 'in whieh these equations (26) and (27) may be expected to 

 assume simple forms. 



Case 1. When the coils are so far apart that M 2 is negligible 

 compared with LZ. ^ 



In this case X^=0, a-t-/3= 7 , a— /3=y- j an d it will be 



found that the expression for the primary current i as given 

 : in equation (26) reduces to the expression (4), which we found 



