298 DE. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
between X 0 and X R , # &o. Then since the equations of transformation (56) are linear, 
any term UfVi m Wi"U/V/W/ in Q 1 (F 1 + F 2 ) transforms into the sum of a number of 
terms X 0 a Y 0 6 Z 0 c X/Y R e Z/ such that 
l+p = a + d, m+q — b + e, n + r = c+J. 
This is not true in the case of Q\ (Fi + F 2 ), since (by 54-) iffi, V'i, Wfi are not rational 
functions of the variables X, Y, Z, but it is true of (Fi + F 2 ) JQ' X de, since the integration 
with respect to e causes all the irrational terms in Qfi to disappear.! This may be 
proved quite generally, but it will be sufficient here to indicate the proof for the case 
Qi — UiCh! s being any positive integer. We may write 
where 
Q\ = UhCW = (X + aC R sin 0 R cos e + 0 o ) (C 2 + 2aC 0 C R sin 0 O cos e) s , 
& = /Di sin Xl2 , X = Xq + M 21 ^dC R COS Xl2 , 
C 2 = C 0 2 + M21 C R 2 +2 / x 2 ! /a (X 0 X R + Y 0 Y e + Z 0 Z r ) COS Xl2 , 
so that X is of the first degree in X 0 or X R , and C 2 is of even degree in the variables 
(X 0 , X R ), (Y 0 , Y r ), (Z 0 , Z h ). The only terms in Q\ which do not vanish on integration 
with respect to e are of the form 
X { s C 2p (C 2 ) s 2p (2aC 0 C R sin 6 0 cos e ) 2p } 
or 
(C 0 C R 2 sin 9 0 sin 0 R cos 2p+1 e cos e + <p 0 ) {2a 2 s C 2p+1 (C 2 ) 5- ^ -1 (2aC 0 C R sin 0 o ) 2p }. 
Now we have 
(C„C R sin e,,) 2 = C 0 2 C R 2 ( 1 —cos 2 e 0 ) = {C 0 2 c r 2 -(x 0 x r +y 0 y r +z„z r ) 2 }, 
which is an even function of X, Y, Z, and can be included under the symbol C 2 . 
Thus, on integration with respect to e, the above expressions become (apart from 
a factor not involving X, Y, Z explicitly) 
& 
XC" S , (C 0 C R 2 sin 6 0 sin 0 K cos </> 0 ) C 2(s_1) 
and by (50) the latter may be written 
C 0 C R 2 (cos X — cos 6 0 COS 0 r ) C 2(s_1> 
[X 0 C r 2 -X r (X 0 X r +Y 0 Y r + Z,Z r )] C 2( *W 
Both these expressions, and consequently JQi (ifii, V\, W\) de as a whole, are of the 
form XC 2s in the sense above defined. Similarly it may be shown that jllVCW de is 
even in all three variables (X 0 , X R ), (Y 0 , Y r ), (Z 0 , Z r ).- 
* So that, for instance, x 0 2 , x 0 x B and x R 2 will all be regarded as even functions of x. 
f The explicit occurrence of x, y, z in J Q'j ch is here referred to ; the latter may involve C R irrationally 
through xi 2 - * 
