292 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
The Motion Relative to the Mass-centre. 
§ 4 (D) Relative to these axes the molecules are initially moving along parallel lines 
with equal and opposite momenta ±m 0 (uiP^'Cr, by (39). The plane containing these 
two lines is clearly the plane in which the inter-action and motion of the molecules 
will take place during the encounter. It is parallel to c R , but its orientation e about 
this direction is independent of* c 0 , c R , i.e., it is one of the additional variables needed 
to specify the encounter, and, similarly, so also is the perpendicular distance p between 
the initial lines of relative motion. It is convenient to measure e from the plane 
containing c 0 and c R . 
In the plane of relative motion so defined, the molecules describe orbits which are 
similar to one another (the origin C4 being the centre of similitude), and symmetrical 
about the line of apses (i.e., points of closest approach). Each orbit has two 
asymptotes, one being the initial, the other the final line of motion ; the distance 
between the pair of final asymptotes is clearly equal to that, p, between the initial 
asymptotes. The angle y ]2 between the two asymptotes of either orbit measures the 
deflection of the relative motion due to the encounter; for molecules of given types 
it is a function of p and C R # only, the nature of the function depending on the law of 
inter-act-ion between a molecule m x and a molecule m 2 . We shall find it convenient, 
for the sake of generality as well as of brevity, to retain y 12 as an unspecified function 
of p and C R in our equations ; the special properties of the molecules under 
consideration are, throughout our work, involved only through the dependence of 
Xi 2 on p and C R . 
It is easy to see that the magnitude of the relative velocity (nin^-Qn is unaltered 
by the encounter, i.e., 
(44) C K = C' R : 
for by the equation of energy we have 
(45) \ (mA’+mA 2 ) = i(m 1 C?+m 2 CV) = (C 0 2 + C R 2 ) = £w 0 (C 0 2 +C' K 2 ) 
by (42) and (43). 
The Velocities in Spherical Polar Co-ordinates. 
§ 4 (E) The above analysis of a molecular encounter may be made clearer by the 
following figure, in which x, y, z, c 0 , c R , c' H are the points in which a unit sphere 
* That is, on p and on the amplitude C R of the vector c K . 
