294 Journal of the Asiatic Society of Bengal. [N.S., XVII, 
od 
(a) Linear oscillator. 
The equation of motion is t+px=0; .. 2 + px*=const. 
; ; 2 
= 2H (twice the energy) =a” when a is the amplitude. 7T’= arg 
to variation from one oscillator to another. e therefore 
have 7(H,—H,) =nh orif H,=0,TH,=nh. In the usual nota- 
tion «, (the energy) = m= ahy (where v is the frequency). 
(6) Rotator. The integral of energy is }Jw*=H where J 
is the moment of inertia and w the angular velocity. 


pa 2 tv 2d 
® H? 
\ ran a/2J .2.H=nh; if H,=90, 
0 0 
272 
we have <_< 
a 82 J 
TWO DEGREES OF FREEDOM. THE -ORDINARY NEWTONIAN 
ELLIPsE. 
We may write the quanta condition in the forms 
[pa .6H=nh and [$apdpy=n'h 
or in the alternative forms 
\ TsH=nh and \ Ddp =wnh, 
0 
where p, is the angular momentum and @ is the azimuthal 
period which, in the present case, is equal to 27. 
Th “ cs - eco “ 
e energy H= —H=} { m(r +76 y=} and the angu 
r 
lar momentum = p= mr’ >. 
The periodic time 7 is obviously a function entirely of the 
energy because both are functions of the major axis. It may 
however be conveniently found directly in terms of the energy: 
a St Senin iy 

