Electron Theory of Matter. 183 
m=l + 2k, while in A... we nave l = 2jn, and in A... 
l — 2s + l. The terms in A ... corresponding to ^=0 vanish, 
and the largest terms left are those corresponding to j= + 1, 
which are of the orders m = + 2n + 2k, and very small. We 
may neglect A ..., as well as the quantities B , U ; this only 
amounts to neglecting the radiation, which in the present 
case is of the same order of smallness as the steady motion 
radiation from the ring. For the same reason we shall 
neglect the small terms involving the variation of mass, m, of 
the electron in equations (12). 
A second simplification occurs in any case of periodic 
motion, whether damped or not, and therefore also in the 
present problem. The terms due to radiation on the left- 
e 2 
hand sides of equations (12), that is, those involving - 8 
P 
as a factor, are easily seen to cancel, term by term, corre- 
sponding terms on the right-hand sides of equations (14). 
Hence in (12) we have only to take into account the 
following terms : 
fjL%—(fJL + m)(Dri, (n + m)wt; + mTi—fjL(o 2 7}, mt, . (16) 
on the left-hand sides ; while on the right-hand sides of (14) 
we need only take into account the real parts of the factors 
of f and at? in the first, of it; and 77 in the second, and of £ 
in the third equation. 
§ 17. We shall now apply our results to our first problem, 
the calculation of the part of 8m due to induction, that is, to 
the axial magnetic force h cos 0, w'hich produces the radial 
mechanical force — fteh cos (6), and while it is changing, 
the tangential induced force <?E = — ~~ -j- (h cos 0) (11). 
When everything has become steady, 8111!= (0, 0, hie8(J3p)) 
(§ 9. (b) (8)). Now we have 
8/3 = ? , S P = -77, • whence 8 (j3p) =/3(^ -y\ 
HenCe Sm^lnefll^-q) along Oz. . . (17) 
This equation shows that, although the variations of the 
h 
field are comparatively slow, nevertheless - is of the same 
order of magnitude as 77 and f. w 
In the present case we have k = ; hence by § 15, 
H = M = N = 0. 
