Systems with Propagated Coupling. 437 



-vibrations o£ the membrane, say n , and therefore 



</h: 



= n 0\/~-^-T- 



V it* 



^vvhere t e — time-constant for electrical circuit, 



t = „ „ „ „ membrane ; 



in each case when isolated from the o£her vibrating systems. 

 If £ T is more negative than (— g) the vibration will grow, 

 whilst the equation for y is then 



Z/=-|^. -fF^-^+^+a'e?cos(B^-.V), 



where P = positive quantity. 

 The value of W is given by 



TV 2 , A2_ .1 . 



J/ + 2A' 



hence for a small positive value of A it is less th&nj/g and 

 will always be less than the free frequency of the membrane. 

 For all positive values of A, y will increase up to a value at 

 which the assumed equations are no longer valid owing to 

 the higher order terms (which have been ignored) acquiring 

 increasing importance and ultimately controlling thp maxi- 

 mum attainable value. 



So far the time of propagation of the .mutual action has 

 been taken as zero ; the change necessary to allow for it is, 

 as in the previous example, to introduce into the value of a. 

 •a factor e~n x . 



The auxiliary equation becomes 



or P+gP + hZ+j=0, 



where 7i = \ + / y a 



_ KL + rR — a 2 cos 2^,/' 



and 



sin 2ax 



m\j 



The critical case for which maintenance will begin is 

 Phil. Mag. S. 6. Vol. 41. No. 243. March 1921. 2 G 



