250 Proceedings of the Royal Society of Edinburgh. [Sess. 
We have now to ‘‘quantise” the variables x and y, using the form of 
Wilson and Sommerfeld. ^^pdq = nh, where n is an integer, h is Planck’s 
constant, and the contour C is the ellipse. Here 
= mx = - ma cos , 
Py = my = mb sin ^ . 
For the ellipse, cp varies from 0 to 27 t. 
If and are integers, 
^ Pxdx = mOr‘ 
sin^ cp . <p dcf ) , 
II 
to 
f'llT 
1 COS^ cf) . cf) dcf) . 
Substituting from (2) for q>, we have 
(3) 
n-Ji^ mna^ / sin^ p 
i:- 
+ 2/<e cos y> + e^ cos^ <p) 
1 — COS^ (f) 
I Z 7 T 
nji = 7mib^j cos^ p 
2 .-\/ (1 + 2^6 COS p + cos^ cf)) 
1 - cos^ cf) 
dcf ) , 
dcf) . 
The simple case of one centre of force follows at once by putting 
/X 2 = 0 and /c = l. 
nji= mnd- 
, si 
Jo 
7L.ji = mnad 
sin^ p 
— e cos cf) 
"->fT 
dcf) = 
’jirmria- 
[1 
cos" cf) 
e cos cf) 
dcf) = 
2'n-mnd^ 
vi - - VI “ > 
which shows that all values of e are not possible, e being restricted to 
satisfy the equation 
V 1 — e^ = 1 — (?^2Vi)^- 
The elimination of e between the equation for nji and nji gives 
and substituting for a in 
we obtain 
(n^ + n2)li = 27tV 
E=. -H = df 
2a 
g _ ^TT^phn 
(tZj + 7 i 2 )Vd 
The form given by Sommerfeld follows at once, viz. 
E-E' 
he 27r^/udm 
X ^ 
1 
Vi + «2P K'+Vf 
