30 PROCEEDINGS OP SECTION A. 



Hence if 



„ a& „ ah .J. 



s s - c 



tlie differential equation is reduced to ^ 



(■P- + ^1^) {D' + u,2.^) cp^O, (II) 



whicli shows that the resultant motion is that due to two superposed 

 harmonic motions whose individual frequencies are wi and wo. 



It will be seen in the sequel that the above method of presenting 

 the solution for the resultant frequencies by the aid of a triangle (which 

 we shall call the triangle) simplifies many of the considerations relating 

 to coupled circuits. It at once enables us to follow the variations in 

 the resultant frequencies due to variations either of the coupling or of 

 the natural frequencies of the two circuits. Thus if the natural fre- 

 quencies of the circuits be constant while the coupling varies from very 

 loose coupling (when ^ is small) to very tight coupling (when \p 

 approximates to a rigiit angle and consequently the angle 2\p included 

 between a and b becomes iiearly 180°) the squares of the reciprocals 

 of the frequencies are given by 



1 _ a + 6 c 

 i72 2ab' - 2ab 



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

 to a + 6 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 r, for any 

 value sin \p of the coupling, are always given by the formula 



= ^/2 t sin 



V 



(i-^l) 



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 : — 



C\ = Ax cos wit + Bi sin wit + Pi cos W2.t + Qi sin u^t 

 Cz = Ai cos Wit + B2 sin wit + P2 cos w^t + Q% sin w^t 

 where Ai, Bi, &c., are constants to be determined. 



But Ci and C2 are connected by the equation (see Eq. (I) § 2) 

 Li (Z)2 + a)Ci + MD^Cz = 

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

 Hence 



A2 _ B2 __ Li (a - w]2) _ Zi s - b 

 Ai~ B'i~ MZ^ M ' 'T" 



P2 _ Q% __ Li (a — 0)2^) _ _ Li s -a 



