298 Journal of the Asiatic Society of Bengal. [N.S., XVIII, 
It is now obvious that instead of associating a quanta 
number with the phase-integral J Té6H we may associate another 
1 
with the integral P, ia, | which will be the difference of the 
1 
quanta numbers associated with the former and with the ex- 
pression 2rp, and this is exactly what Sommerfeld has done. 
CASE OF THE RELATIVISTIC ELLIPSE. 
Phase-integrals :— 
[pa.sH=mh; [fagsp,=n'h 
Sommerfeld writes 
£ (mz) =— cos ; “( j)=-© sin ? 
wt «5 ior” tnadk J sop 
where m=mo// 1 — BP and B=- 
Putting K = — mc? 1— BP we have OR Pe Bee ; L=mz 
dx a/1—P 
and oe 
oy 
The equations of motion are therefore deducible from 
F 2 
) | (K~V)dt=0, where V is the potential -= . 
The angular momentum sions a .Pb6=mrg=p. 
a6. 4/1 —# 
The integral of energy re be found oct 
sf V) dt= \ Gia) 
Regarding upper limit variable and 4 a variation along the 
orbit we hav 

-OK OK 
K—V=x — +4 —-—const. (energy), 
dx Yay 
-OK OK 
or A= Pa 
=" os + Voy K+V 
m c e 


evieg F. 


