254 Prof. Barton and Miss Browning on Coupled 



Inserting these values in (25) and (26) they may be 

 written : 



p ffy P + Q-fffQ v g /3 PQ g * 



r dt* ± (l + (3)(F + Q) r TV- (1 + /3)(P + Q)1 *> ' KLi) 



Q d*z P + /3P + Q 'g __ g PQ </ . . 



*#■ + (l + ^)(P + Q)^7 -(l + y 8)(P + Q)^- * ^ 



Comparing (27) and (28) with the electrical equations (1) 

 and (2) we see that in our present mechanical case the 

 analogy is not exact. For the mutual induction formed the 

 coefficient of a second differential coefficient of quantity of 

 electricity, whereas in our analogue the cross-connecting 

 factor which replaces mutual induction is a coefficient of the 

 displacement itself which is taken to represent the quantity of 

 electricity. 



This naturally suggests the question as to how the 

 coefficient of coupling is to be estimated in this mechanical 

 case. If this coefficient is to remain a pure number as in the 

 electrical case, the answer is clear. For we must take the 

 product of the coefficients on the right side of the equations 

 and divide by the products of the coefficients of like terms on 

 the left sides. 



We accordingly obtain 



_.2_ P. £_j (2Q) 



7- (P + Q+/3Q)(P + /3P+-Q)- • ' v* > 



For the simpler special case where P = Q that will hence- 

 forth be dealt with, (27) and (28) may be written 



(2 + 2/3)J + (2+/3)m 2 y=/3m^ . . . (30) 



lH 2 Z 



(2 + 2/3)^+(2+/3)mh = /3m% . . . (31) 



where m 2 is written for g/l. For this case the coupling 

 becomes 



y=2T/3 (32) 



Solution and Frequencies for Equal Bobs. — On putting 

 in (30) 



*/ = **', (33) 



we find 



, m Q±Mte£f±it*A . . . (34) 



j3m 2 



