46 PROCEEDIKGS OP SECTION A, 



when the second power of the damping coefficients is neglected are 

 wi and W2 respectively, and Wi, ^2,^1, u-z are determined by the 

 above relations in terms of the constants of the individual circuits of 

 the transmitter and of their coupling. 



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

 circuits are equal, it is easy to show that 



ni=-^ (Xi + X^)= "^{X, +X2) 



n, = ^^ " ^ (Ai + X2) ^ ^!(Ai + A2) 

 ^(s - c) 2a 



hence -J- = — ^■ 



W2 W2 



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

 as wj is less than w^, 



^ being equal to ~ ^ , 

 W2 s 



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

 damped. 



The same is true in general, as can be seen by putting Eqns. (Ill) 

 § 16 in the form 



2 % cos ^ i/' = Xi + A2 - (Ai cos B 4- A2 cos A) 

 2 n2 cos 2 ^ = \j + X2 + (\j cos 5 + A2 cos A) 



18. The general solution of 



(D2 + 2niD + a;i2)(Z)-2 + 2n2D + ^2^) = 

 is of the form 



= Axe~ ' cos (j^xt + xi) + -^26 " cos (w2^ + X2) 

 where A^t A^-, xij X2 ^.re constants ; and <\> may represent Fj, F2, 

 Ci, (72, the constants having different values for each. 



In order to obtain the complete solution for given initial conditions 

 use must be made in the same way as in § 5 of one of the four relations 

 similar to 



(i)2 + 2AiZ) + a) Fi = - piD^Vz 

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



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

 into account, and if when t = 0, Vi = E, V^ = 0, Ci = 0, C2 = 0, 

 that the complete solution is given by 



7j =r - j (s - a)e ""I'cos {w^t + oj) + {s - 6)e~"'2' cos (w2^ + 02) [ 



E \ -n t _ ) 



F7 = Pi a - j e 'cos (wi« + fti) - e "■!' cos {w2,t = ftz) \ 



