Structure of the Atom. 243 




Again, 
or 
» Ne) Cos a ey 
bla 2 mer n Aka n 
oB, +B, +o" B,_1= ~; ({ sin — ies 
oa n OF: TOD, nhs 
a sin7 
n 
30 
kor we n 
+ sin — —— -— 
ee ie 
sin? — 
; =tMz, say. 
Substituting these values, equation (1) becomes 
((A—mg?) + Lz)(C—mq?— Nz) =(Mz—2maq)?. . (2) 
From the value of C given on p. 240 we see that C is the 
value of N, when k=0, and so may be denoted by No, and 
2 
that A= : - S—L,; hence equation (2) may be written 
“2 S+L,-—L,- mg?) (No—N,—mq’) = (M,—2meq)?. (8) 
kin this equation may have any value from 0 to (n—1); but we 
see that if we write n—£ for k, the values of g given by the two 
equations differ only in sign, and so give the same frequencies; 
thus all the values of g can be got by putting k=0, 1,.. Pcty 
if n be odd, or k=0, 1, = if nm be even; thusifn be odd there 
= : equations of the type (3). When k=0, M,=0, and 
(3) reduces to a quadratic equation; so that the number of 
1 ee ae 
equations 1s 4 x —- -2= eeu nm be 

are 

nr 
roots of these 
even there are ; +1 equations; but as Mz;=0 when s=0 
and k= _ two of these reduce to quadratics; so that the 
. . nr 
number of roots of these equations is 4(5 +1)—4=2n. 
Thus in each case the number of roots is equal to 2n, the 
number of degrees of freedom of the corpuscles in the plane 
of their undisturbed orbit. 
Let us now consider the motion at right angles to this plane. 
By equation y we have 
d*zy ve? 
Te -_= Be zp+ Dip—>DsZp45 ; 
We 
