302 Mr. S. H. Burbury on some Problems 



the position of p is determined by that of G. Hence 

 Tj>=F P f$dZcl V dZ7rs*pf p , 



and since all the directions of the relative velocity before 

 encounter are equally probable, 



in which ff dS denotes integration for all directions of the 

 diameter, P'Cp', of the spheres described about C, or £ rj £ 



through P. Let ^ jjFp/^dS be denoted by ¥ F >fp>, then, 

 given P, 



^=j$4<m?™m*v^-Fp/,) ( 



and by symmetry -^ has the same value. In this equation 



f p - and f p are supposed to be expressed as functions of f 77 ? 

 and the coordinates of P or P'. 



In order to find —j— , we multiply —jj- by log F P and —■ by 



Jog/p, and then integrate over all positions of P in space. If 

 TGp and P'Cp' be any two diameters of the spherical shells de- 

 scribed with radii R and r about C, -7— will contain the term 



' dt 



«V (log F P + log/,) (F P ./,, -Fp/,) ; 

 that is 



7T^log(Fp/,)(Fp,/,,-F P /,). 



By symmetry, as P ; is a position which P will assume in 

 the integration, -=- will also contain the term 



7r^log(Fp./,,)(Fp/,-Fp/,-). 



s tog 



™Vog|^(Fp,/,,-Fp/,), 



and will 'consist wholly of a series of terms of that form. 



Now this expression is necessarily negative unless 



Fp'/p' = F-pfp, and is then zero. Therefore -7- is necessarily 



And adding the two terms together, — — contains the term 



Cut 



