1921-22.] Quantum Theory of Secondary Spectra. 
249 
Let us consider an ellipse described under the action of two central 
forces attracting according to the law of the inverse square of the distance. 
We have 
m \ ?*2 CL J 
where and are the distances of the particle of mass m from the foci 
of an ellipse of semi-major axis a. 
The velocity V required for the description of the ellipse under two 
centres is given bjr 
^ ^ 7ni\ mi\ ma 
H = sum of kinetic and potential energies 
= \m{v^ + Ih , 
Q\ H = — ^ 2 
^ ^ * 2a ■ 
In the case of an electron of charge e, attracted by two positive nuclei 
of charges and K^e, and = K^e^. 
For the ellipse : a? = a- cos <p , ^ = 6 sin <p , <p being the eccentric anomaly, 
y2 ^2 _[. 2/2 ^ ^^2 gq^2 cos'^ 
— a^{l — cos^ <;^>)^^ 
where e is the eccentricity of the orbit. 
Potential energy = 
/^i 
a(l - e cos 
+ 
IH 
a(l -1-ecos </>) 
Hence from (1) we have 
/Xj + /x-2 + e(/xi - /Xg) cos cf) 
a(l - cos‘^ (f>) 
^ma^(l - cos^ -{■ 
+ + e(/xi - /X2) 4^ 
a(l - cos'^ (fi) 
2a 
^ ma^ 
l + 2e^i , COS cf)-{-e^ cos^ cf> 
/^] +/Xg 
(1 - cos''^ (fi)^ 
. cj)^ = . 
1 2k6 cos 4> + e^ cos^ ^ 
(1 - cos^ (j^y 
( 2 ) 
where 
= (/xj -P fx.2)lma^ and k = (/xj - + h) • 
