45 
1919-20.] Note on the Scattering of X-Rays. 
if L GBA be denoted by <p. Equation (1) thus takes the form 
n + 22 cos (2&L cos cos <f>) . . . (2) 
Now BZ, BG, and BQ are fixed directions, but the atom may be 
orientated in any direction whatever with reference to the incident 
light, i.e. all directions of BA are equally possible. We consequently 
require the average of the term under the summation sign in (2) for 
all the different values of <£. The number of times <p has one particular 
value is proportional to the area of the zone comprised between <p and 
<p-\-dxf> on the sphere of unit radius, i.e. to 27 r sin <fid(p. The average is 
therefore 
1 /> 
cos (2&L cos cos </>)27r sin 
r+i 
= cos (2kLx cos \6)dx 
sin (2&L cos \Q) 
~ 2A;L cos 
Substituting this value in (2), and at the same time allowing for the 
variation in the scattering from a single electron, we obtain finally 
I'./ 2 (l + cos 2 6){n+ 2 
sin (2&L cos J0)\ 
JcL cos 
(3) 
for the energy scattered in direction 0. 1^/2 has been distinguished with 
a dash, because it no longer represents the energy radiated in direction J 7 r. 
The latter is now given by 
Itt/2 = I'tt/2 n + ^ 
^ sin cos r ) 
ldL cos \tt 
(4) 
There are \n(n— 1) possible pairs of electrons, i.e. \n{n — 1) terms under 
the summation sign or in the case of lead J82 x 81 = 3821 terms, and before 
the summation can be made we must know the value of L for each of 
these pairs. If the wave-length is very long, k = 0, and 
sin 0\ = + cos 2 Q)(n + n(n -T))l 
0 
(5) 
I = I'^(l + cos2 6)[n + ^2 
= Tw2^ 2 (l + cos 2 6 ) ) 
i.e. the energy radiated is n 2 times greater than the original formula 
would lead us to expect from a single electron, and n times greater 
than it would lead us to expect from the atom. This result has already 
been anticipated. * 
§ 2. In order to study the possibilities of formula (3) it is advantageous 
to take the case of n = 4t and L the same for all six possible pairs of electrons. 
* Barkla and Dunlop, Phil. Mag., xxxi, 1916, p. 231. 
