40 
Proceedings of the Royal Society of Edinburgh. [Sess. 
out from a point source is a mathematical fiction. What we really have 
is a very great number of spherical wavelets, each diverging from a 
different electron, criss-crossing in various directions, and consequently 
interfering with one another. For example, suppose there are n electrons 
in the source, all close together, and that the intensity of radiation is 
required at a point P at a distance r great in comparison with the linear 
dimensions of the source, and so sensibly the same for all the electrons. 
Let the intensity at P due to a single electron be I/r 2 . Then the resultant 
intensity may be anything from 0 to n^I/r 2 , according to the number of 
wavelets coincident in phase at P, the lower values predominating. If the 
phases of all the different waves are absolutely at random, the problem 
reduces to a celebrated one solved by Lord Rayleigh, and the chance of 
a particular intensity J is 
r2 
In 
—Jr 2 /In 
e dJ. 
If there is any regularity of structure in the source, Lord Rayleigh’s 
expression may not do justice to the higher intensities. 
Thus even if all the atoms absorb, they do not absorb to the same 
extent; some absorb much more than others. This follows simply from 
the laws of chance. 
§ 4. Objection may be taken to the friction term in the equation of 
the electron. We cannot speak of frictional forces in connection with 
electrons; the term represents loss of energy into the free vibrations of 
the electrons in a somewhat unsatisfactory manner. 
Let us try to construct an expression for the absorption band without 
using the conventional friction term at all. Suppose that when t = 0 
both x and dx/dt are equal to zero, a reasonable enough supposition, since, 
as the motion is then random, the mean values must be zero, and let 
d 2 x _ Ae , 
+ p% __ 00 sgt 
<») 
represent the motion of the electrons. Then, considering both free and 
forced vibrations and determining the constants from the initial conditions, 
Thus 
f(t) = x = 
- Ae 
m(p 2 — g 2 ) 
(cos gt - cos pt). 
[ f ( t ) cos gt dt = 
Jo 
Ae 
m(p 2 — g 2 ) 
- Ae 
2 m(p 2 — g 
I (p sin pt - g sin gt) cos gt dt 
Jo 
b\( 2 cos 2 gt cos (p + g)t - -P— cos (p - g)t) (10) 
2 )V P + 9 P~9 
