530 Dr. L. Silberstein on Molecular 



At first sight it would seem that, if each atom is a sphere 

 of radius r. we should have -Jo- = -|R -|- r (and in the case of 

 contact, for example, ^a = Ii), assuming, of course, that our 

 optical centres 1, 2 are the centres of the atomic spheres. 

 This relation between R and a, r would hold, in fact, if the 

 system had a sufficiently rapid motion round the mass-centre 

 with chaotically changing axes of rotation ; for then the 

 molecule would behave, for purposes of encounter, as a full 

 sphere of radius ^a = ^R-\-r. But, contrary to the opinion 

 of some physicists, this is by no means the case. If there 

 is any rotation, it is certainly not rapid enough for the 

 rushing neighbour molecules. The radius ^a of the equi- 

 valent sphere is not \R + r but rather (much the same as if 

 the molecule was not rotating at all) equal to the average p 

 of the semidiameters p drawn from the centre of the 

 system to its "surface" (more rigorously, the semidiameter 

 corresponding to the mean area of section of tangential 

 cylinders). We shall not be far from truth if we substitute 

 for the two spheres * an ellipsoid of revolution, of semiaxes 



a=-iR + r, h = r (21) 



The deeper the interpenetration of the atomic spheres the 

 more justified this substitution, but it would be sufficiently 

 accurate even in the case of contact. The corresponding 

 mean ~p = \a is easily found to be 



o=-— F,,cos0=-> .... (22 a) 

 it a 



where Fe is the complete elliptic integral of the first species 



whose modulus is sin 6. Thus, with the semiaxes (21), and 



writing for the moment 



x = 2r/K, 



we shall have, for the ratio of the (free-path) semidiameter 



of the molecule to the interatomic distance i£, 



&V F ''.~'--r£i- • • • ( 22) 



If, for instance, the spheres are in contact (#=1), we have 

 0=60°, and \g : JR= 0*6864. This ratio increases with in- 

 creasing depth of interpenetration of the atomic spheres. 

 In genera], a and It being known, from the kinetic theory 

 and from the present dispersion theory, we have in (22) a 

 transcendental equation for a, and therefore for the atomic 



* The spherical form is assumed for the sake of simplicity only, and 

 can later be given up for a more general one. In fact, the circumstance 

 that (jb (Boyle-diameter) differs from o (free-path) even for monatomic 

 gases would suffice by itself to show that at least some of the single 

 atoms themselves are not spherical, or, that the equivalent force-centres 

 are not isotropic. 



