294 Mi, S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
the expressions for the initial and final velocities and velocity components in terms 
of Co, c E , p and <?, as follows :— 
(51) mi Cj 2 = 0 12 (cos 0 O ), m 2 C 2 2 = 0 21 (cos 0 O ), 
MiC'i 2 = 0 12 (cos 0' o ), 
m 2 OV = 0 2 i (cos 0' 0 ), 
(52) 
j m/' Ui = jU-i'-Gq COS A + ( u 2 1/ ’Cj; cos 0 E , 
= lUi l2 C 0 cos A + m 2 1/2 C e cos 0' e , 
M 2 U 2 — m 2 '"C' u COS A — Mi /2 C e COS 0 e , 
m 2 ' /l U' 2 = nf-C o cos A—mi' -Cj. cos 0' R , 
where we have adopted the convenient notation* defined by 
(53) 
f 0 i 2 (cos 6 ) = miC 0 2 +m 2 C e - + 2 (miM 2 ) 1 / ’C u C h cos 0 , 
[021 (cos 9 ) EE m 2 Qj 2 +miC e 2 —2 (mim 2 ) ^C 0 C e cos 6 . 
Equations (51) to (53) are merely particular cases of (42), (43), expressed in terms 
of amplitudes (51, 53) and of ^-components (52). The latter might also have been 
written in terms of the components of C 0 and C J{ , as, for example, 
(54) \J\ = Xo + m^Xr = X,, + m 2 i ,/ 2 {X R cos Xi 2 + (Y r 2 + Z r 2 ) V2 sin X i 2 cos (e + 0„)}, 
by (49), writing (X 3 , Y 0 , Z 0 ), (X R , Y J{ , Z R ), (X r R , Y E , Z R ), for the components of c U5 c R c' R . 
Equations similar to (53), (54) may easily be written down also for the y and 2 
components of the velocities. 
The Dependence of U^, V'j, W\ on X i 2 - 
§ 4 (G) From (51) and (54) it is clear that any function Q, (U'i, \l\, W'i) of U\, W' 
is a function of Ui, V,, W,, U 2 , V 2 , W 2 , p and e, or of U 1} V;, W ]5 U 2 , V 2 , W 2 , Xl2 and e, 
since p is involved only through Xl2 (though Xl2 is not entirely independent of the 
preceding six variables, since it depends upon C R ). If Qi (U'i, \l\, W'i) be regarded as 
a function of Xl2 , when Xl2 is made equal to zero it reduces to Qi (lh, Vi, Wi) simply: 
this may be seen either from (5l)-(54) or, still more readily, from the figure on p. 293, 
since when Xl2 = 0, c' R becomes identical with c E , and lienee by (42), (43), so also does 
cT with fi. 
Transformation of Co-ordinates. 
§4 (H) In §5 we require the Jacobian of transformation 
j a ( lG, fu w\, 
a(Ui, v„ Wi, u 2 , v.o, w 3 ) 
between the initial and final velocity components, p and e being constant. Since the 
motion during an encounter is reversible, the relation between the two sets of velocity 
* In § 7, for the sake of brevity, we shall write 0i 2 , 0 2 i, 0'i2> ©^i respectively for 0 ]2 (cos 0 O )> ©21 ( cos $o)> 
9i2 (cos 6 ' 0 ), and 0 2 i (cos 8 ' 0 ). 
