Coupled Circuits and Mechanical Analogies, 515 



gravity in the immediate neighbourhood of the origin. It 

 is evident that its kinetic and potential energies are propor- 

 tional to T and U as defined by (5). Hence its motion 

 presents an analogy to the problem considered. The surface 

 is fully defined by the section on the definite plane £=$g~\ 

 and this section is 



(1 - r 2 )P! 2 P 2 2 - P 2 2 f 4- 2 7 . PjP^ + P x V 



= (P 2 ? + 7.P^) 2 + (l-Y 2 )PiV 



=(i-7 2 )ftT+(7.:p 2 f+p^) 2 , 



or clearly an ellipse inscribed in the rectangle £=+I ) li 

 r)= ±P 2 . The ellipse, which is also represented by 



f =Pi cos 0, 7)= — P 2 sin (04- a), sin a = <y, 



indicates by its axes both the periods and the directions of 

 the fundamental oscillations. The area of the ellipse, com- 

 pared with its maximum, is cos a or \/(l — <y 2 ). In the limit 

 the ellipse degenerates into one diagonal of the containing 

 rectangle. The effect of the coupling is very clearly shown 

 by the change in the shape and area of the ellipse, and the 

 slewing round of the axes seems to show very well how 

 the fundamental oscillations must enter into both systems 

 dependently. The two degrees of freedom of the electrical 

 system are represented by the displacements of a single 

 material particle parallel to two fixed rectangular axes, and 

 what may be lost in distinctness seems to be compensated in 

 the very clear connexion between the two displacements. 

 It is the natural type of all small oscillations with two 

 degrees of freedom, and if it is not well adapted to detailed 

 experiments it has the advantage of being most readily 

 apprehended without any experiment at all. 



Fig 



2. 



9 t ^***+*S^^ 



V 



r^> 



V ) 



Fig. 2 shows the corresponding sections of the surfaces 

 when P 1 /P 2 = l , 5 and the coupling y is 0, 05, and 1. 



