Coupled Circuits and Mechanical Analogies. 513 



Or agaiu, since BD represents PiP 2 , and HG represents 

 pijt> 2 , HG being perpendicular to EF, 



0B*= PjXP, 8 + P 2 2 ), BA 2 = P-^Pi 2 + P 2 2 ), 



Hence on the same scale by which P 1? P 2 are indicated by 

 OB, BA, EH represents p x and HF represents p 2 . Thus if 

 OB, BA represent the natural periods, the periods under 

 coupling are shown by EH, HF and their changes can be 

 traced in a much more instructive way than the equations 

 alone, simple as they are, naturally suggest. 



4. There is another elementary point. If #, y are con- 

 sidered as coordinates the fundamental vibrations are per- 

 pendicular to the lines x — \ 1 y = 0, x — \ 2 y = 0, which are 

 not at right angles. Let 



mX^/S = \/R 

 so that (1) becomes 



M v /(ES)-m(NS-LR)-m 2 Mv / (RS)=0, 

 or 



2m _ 2 7v /(LNRS) _ y . PiP 2 

 l_ m 2- ns-LR ~i(P 2 2 -Pi 2 )' 



In fig. 1 this fraction is represented by — GD/CD, and 

 thus m = tan JGOD =tan FOC. The lines 



y = m^v, y = m 2 x 



are therefore represented in direction by OF, OE. But 

 the required lines are 



and the latter are easily derived from the former when the 

 ratio R/S is given. They are represented by OX, OY in 

 the figure. The simple oscillations are perpendicular to 

 OX, OY, and their periods are 2tt/?i, 2irp 2 . The resultant 

 compound oscillations are to be found from their components 

 perpendicular to DA, DB'. 



The fact that OX, OY are not rectangular axes is an 

 inconvenience which can be avoided by taking other vari- 

 ables f, 7] such that 



*=B*ft y=&n (4) 



As this change of variables is independent of the coupling, 

 it simplifies the problem without involving any complica- 

 tion, and can therefore be introduced with advantage at the 

 beginning. The problem is then brought to its simplest 

 terms as the projection of two rectangular harmonic motions 



