514 Prof. H. C. Plunimer on the Action of 



on two other rectangular axes. The fundamental periods 

 are shown graphically in fig. 1, and the inclination o£ the 

 two sets of axes, which is now simply AOF. 



5. The problem may also be considered in a slightly less 

 elementary way. The fundamental equations may be written 

 in the form 



(LN-M 2 )RS:* =-NS^-M%, 



(LN-M 2 )KSj/ =-MS#-LBy. 



Now these equations will also follow in a dynamical problem 

 defined by the kinetic energy T and the potential energy U, 

 where 



2T=(LN-M 2 )(S^ 2 + R2/ 2 ), 



2U=^^ 2 + 2M^+^ 2 . 



The common conjugate diameters of the ellipses 2U = 1, 

 S^ 2 -f% 2 = l, are 



MS^ 2 -(NS-LR>z/-M% 2 = 0, 



or (tf-XtfXtf— \j#) = 0, 



by (1). Normal coordinates are obtained by taking these as 

 axes. But it is simpler to take f, t] as the variables accord- 

 ing to (4) . Then 



2T = (LN-M 2 )RS(P + ^ 2 ), 



2U = NSf 4 2MR*S*£i7 + LP?? 2 . 



Since P 1 2 = LR, P 2 2 = NS, M 2 = 7 2 . LN, these may be 

 written 



2T = (l- 7 2 )P 1 2 P 2 Vp + ^) -, 



2U = P 2 2 f 2 + 27-PiP 2 |'7 + PiVj " ' " * K 



The axes of U are 



7.P,P 2 (r-^ 2 ) = (P 2 2 -Pi 2 )fr, 

 and if these are taken as the axes of X, Y, 



2T= Pl V(X 2 + Y 2 ), 

 2U=p 2 2 X 2 +p 1 2 Y 2 , 



where K+pi 2 =P 2 2 + Pi 2 , Pi 2 p 2 2 = Pi 2 P 2 2 (l-Y 2 J, as before, 

 by simply comparing the last two forms of U. The rest of 

 the solution follows immediately. 



6. The obvious analogy suggested by (5) may just be 

 worth noticing. On the surface 



2(1 - 7 2 )P! 2 P 2 V?= P/| 2 + 2 7 . PiP.fr + Pi V, 

 the axis of f being vertical, a particle is oscillating under 



