Lord Rayleigh on the Induction- Coil. 589 



(no current), when £ = 0, is* 



~1 e~ l> 



■m-t') sin n f {t-t') . U dt'. . . (11) 

 16 J o 

 where 



n'=*/(7i 2 -£« 2 ) (12) 



The various elements of (11) represent in fact the effects at 

 time t of the velocities U dt' communicated {t — t') earlier. 

 In the present case we are to suppose that U is positive 

 throughout, and that \ U dt' is given. 



The integral simplifies in the case of /e = 0, that is of 

 evanescent secondary resistance. We have then n'=n, and 



= - C smn(t-t').TJ 



"Jo 



dt'. . . . (13) 



It is easy to see that the integral, representing the potential 

 at the secondary terminals, is a maximum when U is concen- 

 trated at some one time t', and t is such that 



sinn (t — t') = l, 



that is, when the break is absolutely sudden and the time 

 considered is one quarter period later. If the break be not 

 sudden, sin n(t — t') will depart from its maximum value 

 during part of the range of integration, and the highest 

 possible value of u will not be attained. 



The theory is substantially the same if k be finite. There 

 is some value of (t —t') for which 



<ri*e-*0si nw '(*-Y) 



is a maximum; and the greatest value of u will be arrived at 

 by concentrating U at some time t\ and by so choosing 

 t that (£—0 has the value above defined. The conclusion is 

 that if the primary current fall to zero from its maximum 

 value without oscillation, the potential at the secondary ter- 

 minals will be greatest when this fall is absolutely sudden, 

 and that this greatest value begins to be sensibly departed 

 from when the break occupies a time comparable with one of 

 the time-constants of the secondary circuit. In the case of 

 no resistance we have to deal merely with the time of 

 secondary oscillation ; but if the resistance is high, the other 

 time-constant, N/S, may be the smaller (see equation (2)). 



It is here that the character of the secondary coil, especially 

 as regards the number of its windings, enters into the question. 

 * 'Theory ff Sound/ .vol. i. § 66. 



