1921-22.] 
Quantum Theory of Secondary Spectra. 
253 
where 
/'sm-U/A:V2 
/. ( 
and 
E(?/;, li) - 
j'sin~h/k'\/2^ 
JO 
d(f) 
fl — sin‘^ cfi)^ ’ 
(1 sin2 , 
The spectral lines arising from the orbit about two centres of force will 
be obtained by substituting for a from equation 7 in 
El- 
T. 16E2.2A27t2 
El - 0 7 ^ ~ 
R 
where 
(9) . 
7r2A:'2(7q + n^li^ {n-^ + ’ 
2m7T%^ T -Q IGE^K^ 
N = ^ and R = — — — 
ch^ 
R 
R' 
where n-^ and are integers, and R' may equal R or correspond to a 
different value of the modulus k which determines the eccentricity of the 
orbit. This eccentricity, as in the simple case, cannot assume all possible 
values, but only those values which satisfy the equation determining k, 
viz. equation (8). 
Substituting n^-\-n^ = njS in (9), we have 
( 10 ) . 
l_^rR8^_Pd8'2- 
A __ ?q2 J 
If then we assign a value for d, equation (8) fixes the corresponding 
value of k, and hence R, which is a function of k, is determined. The 
value of S also restricts the values possible for For, if S=plq, where 
p and q have no common factor, (q — p)n^ = pv^, i.e., is a multiple of p. 
Calculation. 
We shall put k = sin 6, Jc = cos 0. Equation (9) becomes 
8 = cot2eil _ V2S^{|[E(w, A')] + E(«, U) 
On account of the values assumed by cot^ 0 between 0 = 0'' and 0 = 24° 
approx., S is greater than unity in this range. The upper limit <p of the 
integrals F('u;, k') and E(^f;, k') is determined by sin <p= <p becomes 
imaginary when A/2cosd<l, i.e., d>45°. When fi = 45°, k'^ = \ and e=\ 
