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



considered as a simple quadratic function, we find that 

 _ — (a + b)± ^(a + bf — 4a5cos 2 ^ 



~" 2 COS 2 l/r 



— {a + b)± <Sa 2 + b 2 -~2ab cos 2^r 

 2 COS 2 t/t 



If, now, a and b be taken as the two sides of a triangle 

 whose included angle is 2-^, then as 



. /s(s — c) 



where c is the third side and a + b + c = 2s, the roots of the 

 quadratic are equal to 



5 — c ah 



and 



Hence if 



COS 2 i|r 





S 



s 



= 



ab 



cos 2 yjr 



s — c 



2 _ab 



r«> 



2 a5 



w i =T' w 2 =; — > .... (I.) 



s — c 



the differential equation is reduced to 



(D 2 + a Jl 2 )(D 2 + a) 2 2 )^, = 0, . . . (II.) 



which shows that the resultant motion is that due to two 

 superposed harmonic motions whose individual frequencies 

 are ft>j and co 2 . 



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

 presenting the solution for the resultant frequencies by 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 frequencies of the circuits be constant 

 while the coupling varies from very loose coupling when 

 ty is small to very tight coupling when i/r approximates to a 

 right angle, and consequently the angle 2-^r included between 

 a and b becomes nearly 180°, the squares of the reciprocals 

 of the frequencies are given by 



1 a + b c 



co- 



2ab ' 2ab' > 



