134 
DE. S. CHAPMAN ON THE KINETIC THEOEY OF A COMPOSITE 
Finally we must define the functions <p' c (y), ( y ), f (t/) 5 these are the only 
quantities in the expressions for p (r, s) and p (r, s ) which depend on the law of inter¬ 
action between molecules. If two molecules m l} rn 2 encounter one another, the 
direction of their relative velocity will he changed through a certain angle 0 12 in a 
plane parallel to the initial relative velocity and to the perpendicular (of length p) 
between their initial and final lines of undisturbed rectilinear motion ; denoting the 
magnitude of the initial or final relative velocity by 
mi + m 2 y/ ^ p e a function 
hm l m 2 / 
of p and y, the nature of the function being determined by the mode of inter-action 
betAveen molecules in proximity to one another. Then Ave have 
(4T7) (pi 2 k (y) = {2k+l){y. 1 y 2 h{m 1 + m 2 )} 'V {l-P*(cos 0 12 )}p dp, 
Jo 
(4'18) \Js 12 k (y) = (2k + 1) {fj.ifx.Ji {m x + m^)}~ l2 y j (I — cos 0 12 )P t (cos 0 12 )p dp, 
Jo 
(4'19) xi 2 k {l/) = (2& + l) {p-ipJi (mj + rn,)}- 1 ' 2 ;!/ j (l — cos 9j 2 ) 2 P /i (cos 6 12 )p dp, 
Jo 
where P ; . (cos @ 12 ) denotes, as usual, the Legendre function of cos 0 12 of order Tc. By 
changing the suffix 1 or 2 throughout into 2 or 1 respectively, we obtain the 
corresponding expressions for <p 22 or <j> n k , and so on ; mere interchange of the suffixes 
does not affect the functions. 
By means of the recurrence formula for the Legendre functions, viz., 
(4-20) 
(&+l)P A+1 — (2k + l) cos @ 12 P^. + /cP^_ 1 — 0, 
we may express \fs and x in terms of the function <p. Thus for \{/ ( y ) we have 
(4-21) 
(y) = 
k 
2k-l 
<t> k l (y)- 
In this way we may proA 7 e that 
(4'22) (y) = (y), x ° (y) = U 1 {y)~ tW 2 (y), (y) = {y) + W (y)- 
From the symmetry, with respect to r and s, of the expressions on the right hand 
of equations (4*07), (4'08), (4'10), (4'll) it is clear that 
(4 2o) Pn ( r i s i ) = Pni ^ d ' i )) P n ( r i ^ i ) = P n ( s i r i )> Pis(^’i^i) ~ P 12> P 12(^T^i) = Z 3 12 (^i 7 *i)* 
(c) Special Values of p (r, s) and p (r, s ). 
To facilitate the exposition of subsequent parts of the work it is convenient at this 
stage to Avrite down certain special cases of the equations (4'07)-(4'12), after executing 
the integrations with respect to x and y. Owing to the generality of the functions 
