Coupled Circuits used in Wireless Telegraphy. 573 



c gradually changing from a—b, its value for infinitely loose 

 coupling, to a + b, its value for infinitely tight coupling. 



Again, if the natural periods of the circuits are equal, the 

 triangle is isosceles, and it is easy to show that the resultant 

 periods t for any value sin yjr of the coupling are always 

 given by the formula 



t = y/2 t sin I -, ± -^ 



where t is the natural period of either circuit. 



Similar statements will obviously apply to the coupled 

 pendulums. 



5. Dealing with the electrical system, the general solution 

 of equation II. § 4 for the currents is 



G 1 = A x cos corf + B x sin coit + P 2 cos co 2 t + Qi sin co 2 t 

 C 2 = A 2 cos corf + Bo sin atf -f P 2 cos w $ + Q2 sm w $-> 



where A l5 B 1? &c, are constants to be determined. But Ci 

 and C 2 are connected by the equation (see I. § 2) 



L 1 (D 2 +a)G 1 + M.D 2 C 2 =0, 



in which a, one side of the triangle, represents ^ as 

 before. 

 Hence 



A 2 



B 2 



L x (a — co 



2 ) 



Lj 5 



-b 





Ar 



"B, 



Mft)! 2 





"M * 



b ' 





p, 



Q 2 



~Li(a — <w s 



2 ) 



L, 



s — 



a 



pr 



"Qi 



Mco 2 * 





M 



' b 





Substituting from these equations for A 2 , B 2 , P 2 , and Q 2 in 

 the expression for C 2 , we find that 



6M" 



®2=T^i (s — b)( A ± cos co^ i-B 1 sin co 1 t)^-(s — a)(F 1 cos (o 2 t-\-Q l sin o 2 t) L 



Now as C 3 = C 2 = when £ =0, it follows that 



A 1= :0 and P 1 = 0, 



hence 



Cj = B x sin g)^ + Q : sin co 2 t 



® 2 ~o~M.\ ( 5 "~^) B i smft) i^— ( 5 ~ a)Q,sina) 2 i V 



