Coupled Circuits used in Wireless Telegraphy. 591 



17. When a = b, that is, when the natural frequencies of 

 the two circuits are equal, it is easy to show that 



a ,_ CO 



Ul = 2s ( l + 2 > = 2ol ^ Xl + **)' 



2 



hence 



n 2 co 2 2 ' 



As the product of wave-length into frequency is constant, 

 and as <»]_ is less than o> 2 , 



— ^ being equal to — — , 



we see that the oscillation with greatest wave-length is the 

 least damped. 



The same is true- in general, as can be seen by putting 

 equations III. § 16 in the form 



2n x cos 2 i|r = X x + \ 2 — (\ x cos B + \ 2 cos A) , 

 2n 2 cos 2 \fr = \ x + X 2 + (X x cos B + \ 2 cos A) . 

 18. The general solution of 



(D 2 + 2nJ) + Wl 2 )(D 2 + 2n 2 J) + co 2 2 )^ = 

 is of the form 



</) = A 1 e-^cos(ft) 1 i + Xi)+A 2 ^-^cos(a) 2 ^ + ^ 2 ), 



where A 1? A 2 , % 1? % 2 are constants ; and cf> may represent 

 Vi, V 2 , Ci, or 2 , the constants having different values for 

 each. 



In order to obtain the complete solution for given initial 

 conditions, use must be made of one of the four relations, 

 similar to 



(D 2 + 2\!D + a)Y 1 = -bD 2 Y 2 , 



obtainable from the early part of this paper (see §§ 1, 2, 3) 

 in the same way as in § 5. 



It will be found, if the first power only of the damping is 

 taken account of, and if when £ = 0, Vx = E, V 2 = 0, ^ = 0, 





