PROCEEDINGS OF SECTION A. 



45 



16. When damping is taken into account, the different variables 

 in the case of the coupled circuits of a transmitter satisfy the following 

 differential equation (see § 2) : — 



{gos2 v//Z)^ + 2(Xi + X2)D^ + {a + b + 4:\^X2)D^ + 2{h\i + aX2)D + ab]<p = 



(I) 

 where a = /<i2^ jf, _ ^^2 sjj^2 j^ ^ 



XlZ-2 



«i 



We will now proceed to investigate this case when squares and 

 higher powers of the damping are neglected. 



Let the operator in (I) above, after neglecting 4 XiX^, be identified 

 with 



cos2 4,(1)2 + 2niD + Wi2)(Z)2 + 9^,2) + ^22) 

 when we jfind that 



cos2 \p (wi2 + 0/22) ~ a + h 

 cos2 -^ wi2(^2^ = ab 



C0S2 i// (Wi + W2) = Xi + ^2 



cos2 \P {w^^ni + wi2«2) = hXi + aX2 > 



The first two of these relations give m^i and w22, and show that the 

 values of the latter in terms of the triangle are the same as before, 

 namely 



„ oh ^ ah 



(11) 



W2^ = 



s - c 



From the last two the resultant damping co-efficients Wj and n-z 

 are at once obtained when we remember the easily-proved relations 

 A „ ^A 



2 ■ "^ """^ 2 



wi2 sin2 — + u)2,^ cos'' 7,- = a 



B 



B 



wj" sin2 - + 0)22 cos2 Z = 6 



where A and B are angles of the triangle. 

 Thus we find that 



ni cos2 1// = Xi sin2 ^ + X2 sin^ — 



■n g 



n2 cos2 \^ = Xi cos2 — -H X2 sin2 - 



Hence Fj, F2, Cj and (72 a^^e each the resultant of two damped 

 harmonic oscillations, whose damping coefficients are nj and W2 

 respectively, and whose frequencies are ^(^^i - n^i, Jto^z ~ ^^2, which 



(III) 



