38 PROCEEDINGS OF SECTION A. 



On the other hand we have seen that the frequency of the resultant 

 oscillations in either circuit is ^ (w2 + wi), and if to be their period we 

 easily find by geometry that 



^' ~ cos ;/'/2 ^ • 

 Hence the number of oscillations in one surge is 



\ = ^ oot I 



which increases and finally becomes infinity as the coupling is loosened. 

 Now the efficiency of the secondary circuit as a radiator depends 

 both on the energy of each oscillation and on the number of oscillations 

 that follow each other in a train, and we have seen that when a — b the 

 maximum amplitude, and hence the average energy of an oscillation 

 during one complete surge is independent of the coupling. Hence the 

 best arrangement is to make the coupling as loose as is consistent 

 with the oscillations attaining the maximum amplitude before damping 

 and radiation have reduced them too much. Thus is explained the 

 advantage of the loose coupling in the Marconi transmitter. 



10. In the general case when the circuits are not tuned to the 

 same natural frequency we have (see § 5) 



E 



Fj = ~ {{s - a) cos M^t + {s -h) cos (o^t], 



which may be transformed to 



p ___^ 



Vi=- J (s -~^^ + {s - 6)2 +2 (s - a){s - b) cos (wg - w^) t 



cos 



== E ^ I - i^^sin^ ^sia^'^^^t COS ( 



where tan ? = tan ~ ~ '^^ t 



c 2 



and 7^ = j)2 -E (cos lo^t - cos w.J), 

 which may be transformed to 



V -OTP /-^l V«6 , . W2-W1 /w2 + wi 7r\ 



^^-^^ v/ ^- -y- • ^'^ ^ «"^ ^- ^cos {^~j- t - -) 



showing that in this case, a^^b, both the amplitude and the phase of the 

 oscillations of Vi vary with the time, and that the amplitude of Vi 



