Electron Theory of Matter. 175 
Equation (1) enables us to assign the upper limit to /3 
(vide table *): for instance, we might admit a secular change 
in the wave-length of the D lines, and therefore of 8, of not 
more than, let us say, six ten-thousandths of an Angstrom 
unit in thirty years, that is, 10~~ 16 of its value per second ; or 
an equivalent change in m, both leading to value of /3 of the 
same order. 
§ 5. The controlling field may be partly due to other rings 
of moving negative electrons, but is mainly due to positive 
charges of such amount as to make the whole system neutral 
except when ionized. The positive charge may be distributed 
throughout the atom (J. J. Thomson's model), or may be 
•concentrated in discrete charges (Nagaoka's model). In any 
case the steady part of the field must be symmetrical about 
the axis of the ring, and derivable from a potential <£; and 
the part of <f> due to any ring can be calculated as if all the 
charges of the ring were uniformly distributed along it and 
revolving with it. In addition the disturbing ring produces 
periodic forces, the fundamental frequency of which is the 
difference between the angular velocities of that ring and the 
one under investigation. These forces excite perturbations 
and cause additional radiation ; they must be small enough 
not to upset the permanence and stability of the system. The 
necessary conditions being supposed satisfied, the periodic 
forces need trouble us no further. Thus we may write 
*-$ ■ » 
§6. At a large distance the ring is equivalent to : — 
(1) a charge ne at the centre ; 
(2) a polarized element of electric moment p = %er ; 
(3) a magnetized element of magnetic moment 
m 
H-'] 
= %% 
r is the vector from the centre to the z'th electron of the ring ; 
v is its velocity relative to the centre, and S denotes sum- 
mation for i from to ra—1. 
When the centre of the ring is in motion with velocity w 
the polarized element (2) is itself equivalent to a magnet of 
moment — p — t- The translator)- motion of the atom as 
a whole may be neglected, even in the case of a gas at high 
* Loc. cit. p. 24. 
t H. A. Lorentz, Math. Encyclopaedie, V. 14, §§ 12-15. 
