400 Sir J. J. Thomson on the Scattering of 



The energy being proportional to the square of the 

 magnetic force is of the form 



—A 2cos 2 — - (vt — (lx r + my t -\r nz r )) 



2tt 



+ 222 cos ^— (vt—(ljc r -\-niy r + nz r )) cos — (vt — (hv 8 -\ my, + nz 9 )) \ , 



where there are n terms under the single summation and 

 n . (n—l)j2 under the double, n being the number of atoms. 

 The average value over a considerable time is 



2^2 [ * + 2 2 2 cos -~ (l(x r —a 9 ) + m(y r -.y s ) + n(z r - z s )) j . 



If x r —x n y r —y s , z r —z s are all small compared with X, 



each of the cosines will be unity and the energy will be 



proportional to 



, n . (n — 1) 2 



n + 2 . — —. , or to n\ 



z 



If the atoms nre arranged in a lattice, so that x r — x s , 

 yr—ys, Zr~z s are always integral multiples of constants 

 a, 6, c, then for certain values of I, m, n the angles will be 

 all multiples of 27r, and the energy in the directions corre- 

 sponding to these values will be proportional to n 2 . If, 

 however, there is not this regular crystalline arrangement of 

 the atoms, but, as in a gas, the atoms are distributed at 

 random, we can easily see that, when the space containing 

 the atoms is bounded by planes, the magnitude of the term 

 involving the cosines is comparable with the square of the 

 number of atoms in a layer whose thickness is proportional 

 to the wave-length, and thus, when the atoms occupy a 

 considerable volume, becomes small compared with n ; so 

 that we may take the scattering due to a gas or a liquid as 

 proportional to the number of atoms. 



From equation (2) we see that if we know the values of 

 a, b, c we can calculate the ratio of the minimum to the 

 maximum intensity of scattered polarized light ; and since 

 we can calculate these values when we know the arrange- 

 ment of the electrons in an atom we have a method, and a 

 very powerful one, of testing any theories of the constitution 

 of the atom ; for Lord Rayleigh has shown that the variations 

 of the intensity of the scattered light in different directions 

 can be determined with considerable accuracy, and that the 

 magnitudes of these variations vary considerably with the 

 nature of the atom. 



We shall therefore proceed to find the values of a, b. e for 



