96 The Law of Distribution [CH. v 



109. Let us now pass to the consideration of impacts in which the two 

 colliding spheres are in given conditions, but the direction-cosines \, p, v 

 may have any values which are consistent with a collision. 



The path described by the centre of the second molecule before collision 

 relatively to that of the first is, since the spheres are loaded, not a straight 

 line ; but since the relative velocity a, /3, 7 is the same for all the impacts 

 now being considered, this path is always the same as regards shape and 

 direction, being in fact a curtate or prolate cycloid according to the magni- 

 tudes of the velocities. 



Imagine the geometrical centre of the first molecule reduced to rest and 

 surrounded by a sphere of radius cr. Let the centre of the second molecule 

 describe its path relatively to the first molecule, then, whenever an impact 

 occurs this centre must lie on the sphere of radius cr. 



FIG. 4. 



Let us start from any point P on this sphere and trace the relative path 

 backwards. This path, being a curved line, either may or may not intersect 

 the sphere again in some point R. 



In the former case an impact at P will only be possible if an impact has 

 occurred while the centre of the second sphere was at some point in PR. 

 In the latter case there is no limitation of this kind, so that the probability 

 of an impact occurring on any small region surrounding P is simply propor- 

 tional to the projection of this area on a plane perpendicular to the direction 

 of the relative velocity at impact. 



Owing to the smallness of r, the relative path will, in general, be only 

 slightly curved. If the maximum curvature of this path is less than that 

 of the sphere of radius cr, then obviously so long as the angle between the 

 normal at P and the relative velocity is acute, the relative path will not 

 intersect the sphere except at P. 



110. For the present we shall assume the velocities of the two molecules 

 to be such that this latter condition is satisfied. In this case the proportion 

 to the total number of collisions of those for which the angle between the 

 directions of the relative velocity and line of centres lies between ty and 

 i/r + dty, is easily seen to be 2 sin -v/r cos tydty, the limits for i/r being and |?r. 



