408 Sir J. J. Thomson on the Scattering of 



or supposing f and r\ to vary as e ipt , we have 



a quadratic equation in p 2 . We have seen, p. 405, that for the 

 equilibrium to be stable for displacements at right angles to 

 the plane of the orbit 1 — 3cos 2 must be positive, where 6 is 

 the angle between the line joining the electron to a positive 

 charge and the normal to the plane of the orbit. Since 

 a/D = sin<9, 1 - 3a 2 /D 2 = 1 - 3 sin 2 6 ; and this is negative if 



1 — 3 cos 2 6 is positive. Hence n 2 — ~ (1 — 3 sin 2 6) must be 



positive, so that the two values of p 2 given by this equation 

 must have opposite signs ; thus one value of p 2 must be 

 negative, so that the steady motion will be unstable. 



Thus if the two electrons repel each other, the system 

 sometimes supposed to represent an atom of helium with 

 two electrons revolving at the same distance from a central 

 positive charge cannot be stable, nor can one when there 

 are two electrons revolving in a circle midway between 

 two positive charges, which is sometimes supposed to 

 represent the constitution of the hydrogen molecule. If we 

 make the extravagant assumption that the two electrons do 

 not repel each other, the problem is the same as that of the 

 single electron already discussed : the steady motion would 

 be stable, and since the relative values of a, b, c would be 

 the same as for a single electron the ratio of the minimum 

 to the maximum intensity of the scattered polarized light 

 would be that given in the table on p. 406. 



If the hydrogen molecule is a system like that represented 

 in fig. 3, with two positive charges and two electrons at the 

 corners of a rhombus, w r e see by considering the equilibrium 

 of one of the positive charges that must be 60° ; so that 

 the ratio of the minimum to the maximum intensity of 

 scattered polarized light would only be *4 per cent. Lord 

 Rayleigh's experiments show, however, that for hydrogen 

 the ratio is at least ten times greater. 



We shall now go on to consider the case when the 

 electrons, instead of revolving in circular orbits, are in 

 equilibrium under forces between the positive charges and 

 the electrons which do not, at the distances which separate 

 the positive charges from the outer ring of electrons, vary 

 inversely as the square of the distance, but at these distances 

 may vanish and change from attractions to repulsions. 



If there is only one electron in the atom, the atom would 



