590 Prof. T. R. Lyle on Mechanical Analogy to the 



Let the operator in (I.) above, after neglecting 4zX{X 2i 

 be identified with 



cos 2 f(D 2 + 2nJ) +co Y 2 ) (D 2 + 2n 2 J) + a> 2 2 ), 



when we find that 



cos 2 ^(ca 1 2 + a) 2 2 ) = a + 6, 1 



cos 2 -^ ft)i 2 o) 2 2 = ab, /Tr N 



cos- -^(?2! 4- w 2 ) = Xi + X 2 , | 

 cos 2 i/r (a> 2 2 ri 1 + ft>i 2 ft 2 ) = 5Xi + a\ 2 . J 



The first two of these relations give co 2 and co 2 2 , and show 

 that the values of the latter in terms of the triangle are the 

 same as before, namely, 



„ ah „ ah 



s s — c 



From the last two the resultant damping coefficients 

 n x and n 2 are at once obtained when we remember the easily 

 proved relations 



A A 



GOi sm J -^ + c«v cos^ — = a, 



B B 



cox 2 sin 2 -^- + ft) 2 2 cos 2 — = b, 



where A and B are angles of the triangle. 

 Thus we find that 



~^ 



\ ■ (in.) 



B A 



n x cos 2 yjr — X 1 sin 2 -^ + ^2 srn2 i> 



n 2 cos 2 -\jr = Xi cos 2 — + X 2 cos 2 -7 . 



— ' -^ 



Hence V 1? V 2 , C 1? and C 2 are each the resultant of two 

 damped harmonic oscillations, whose damping coefficients 

 are n x and n 2 respectively, and whose frequencies are 

 \Zw 2 — n 2 , sJo) 2 —n 2 2 , which, when the second power 

 of the damping coefficients is neglected, are co x and co 2 

 respectively, and w l5 n 2 , &>i, o) 2 are determined by the above 

 relations in terms of the constants of the individual circuits 

 of the transmitter and of their coupling. 



