182 Scientific Proceedings, Royal Dublin Society. 
m is the mass of the electron, and - e its charge, the equation of 
radial motion is 
mr = — eF(r). 
Let a be such that F(a) = 0, then the sphere 7 = a is a position 
of equilibrium. To find the small radial oscillations about this 
sphere, put r = a + #, then 
mé = —eF (a) . 2, 
so that for stable equilibrium /”(a) must be positive. In this case 
the frequency is 
FO 
m 
2a 
Hence, to produce a series of frequencies similar to a spectral 
series, we must find a function F(z) such that for certain zeros a, 
F’ (a) 4q’e'/m = (A + Bn? + Cn*...), 
where n =3,4,5°.... 
A very general solution is obtained by taking /(7) pro- 
portional to 
B. C Ree eaea 
[4 2p oF Bray o I’ (e) sin 272, 
where + = /(v) is an arbitrary function. For 
ene CLAN mi ROks 
A oe FO 
and v = » where n is a natural number, makes #’(7) = 0, so that if 
B= jP), 
Fay 2a Oe ean oa: 
ie 
where / is a constant. If, for example, we put 
QO > OUP ap B ar i” 
Pers 
where a, 3, y . . . are arbitrary constants, we get a function of 
the form 
: Pe 
sin mr{ po+—=+... 
e } 
Di 23 
+ COS Mr (2: + is ar 00 » 
i}? iP 
