318 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES 
We will substitute these values into (166), (167), and write s = 0 , in order to obtain 
expressions for b r0 and c r0 as follows :— 
Cm) 
where we have written 
(172) 
so that 
(173) 
0 ? 
Ko — K — c r o c 0r K r j h 
= 2"'2 r C t K til , 
t = 0 
K ».- = GM“ e-’VWtfdy. 
1 0 7T Jo 
It is of interest to examine also the r th successive difference of 6 r0 or c r0 , which we 
shall denote by S r0 b r0 or S r0 c r0 . We have 
(174) WM =/.( r )-A/(r-l) + ... . 
Then, from (172), it is easy to see that 
r r—m 
S r ^r = 2-' 2 (-2)”VC„ £ r _,C,K tl 
m — 0 t = 0 
= 2- 2 ,C,K t , 2 (-2)-,_,C. 
t = 0 m = 0 
= (- 2 )- Jf-ll'A, 
t = 0 
since 
Hence 
(175) 
r—t 
2 (—2)V,C, = (1-2)'-' = (-1)'-'. 
JA. = = (-2)-'(*„)-’ 2 (-l)' r C,K,,. 
< = 0 
Similarly the r ,lh difference of 
2 Dy 2 r _ s G t Kj i/ 2S+1 , 
t = o 
which is the part of & rs (f-s+l) or c rs (|-s + l) which depends on r (s being even and 
r s) is equal to 
(-WS's (-iy r a.k, 
Z t = o 
Symmetrical Expressions for 2 /3 r and 2 y, 
r = 0 r = 0 
§ 8 (E) While V ( b rs ) and V (c rs ) are symmetrical, the derived determinants 
V r (b rs ), V r (c rs ) are necessarily lacking in symmetry, and our expressions for (3 r and y r , 
when we attempt to make successive numerical approximations to their values 
