316 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
From (96) it is easy to see that 
(156) 
so that 
k\ 
* A ‘ = 
(157) B s+2 (r+ 1, s + 2) 4- 2 - 6 + ^) (r+l, s + l) 
ZS + u 
_ (5 + 2 )! (r+ l)j {r+ %)_t-s-\ r 2 t 2 (r+s + 3 -«) 
(s+f),„ <-». (t+i),(i-»-l)l V 
By putting r in place of (r+l) in (157), and adding {2 (s + 2) y 2 /(2s + 5)} times a 
similar expression in which r, s replace r+l, s + l in (157), we also have 
(158) B-(r, s + 2 )D-A±flrB-+,. + s) 
(s + 2) ! ^ )’t ( ^'T~ir)t-s m2t / ,,2(r+s + 2—t) 
~ uhl; (t+i), («-»)i J 
We now substitute the expressions on the left of (157) and (158) into (154) and 
(155), and integrate with respect to x by means of the well-known formula 
[ e X ‘x 2(t+1) dx = (t + ^) t ', 
Jo 
(159) 
we thus obtain the equations 
(160) 6„(| S + l) = 8B 0 ^'%,fe-='V‘ +a (T2/)4W^ 2 (> ' + V rw ‘-‘ > <fo 
J l S + tb + 2 t = S + 1 (t — S— ljl 
(161) c„++i) = i8C„^v„n-»y«(+ 7 i++- s '>][- r -u i=i t f<’+.±*-'>d y . 
(s + 2)! A (r+ f) t _ 8 „. 2 ( r +s+ 3 ^t) 
(•S + -|)s+2 * = * '(^ — 5 ) • 
or, changing the notation so as to make the lower limit of t zero, and inserting the 
values of \ rs , \' rs according to (125), (126), i.e., 
(162) \ rs = 
we have 
9 —(r + s+4) 
(r+l) (s + l) (r + f) r+2 (s + |-) s+5 
X' = 
2-(i"+ s +^ 
( r + t)r+2 ( S +l")s+2 
(163) 6„(Js+l) 
_ 2-(r+ S +l)g V. _n_ (- g + 2 ) ! [ 
0 (s+l)(r + f), + 2 {(,+*)„,}* 
V +2 (r?/) 2* r _A(r + f) t 2 / 2(r+3 - rt ^ 
t = o 
(164) c„(i* + l) 
= 9.2- (r+s+ 3) aB7T ,/3 —^ 
(s + 2)! 
( r + f)r+2 {(5 + f) S + 2} 2 * 
e~ y2 <p s+2 (ry) 2 r-Alr+fX*/ 2 ^ 3 - 4 ^. 
( = 0 
