306 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
in place of (103); in the first bracket the ambiguous sign is to be + in the case 
0 12 m , and — in that of 0 21 m . 
By the theory of Legendp^e’s functions we have 
(2&+ l) f P,. (cos 0 0 ) P, (cos 6 0 ) d cos 6 0 = 0 if Ic ^ l. 
P* (cos 0 ( ,)} 2 d cos 0 0 = 2. 
Consequently 
(105) 
0 12 ’"0V de d cos 6 0 = 4 tt 2 {2k + 1) m A.\ 2 n A k 12 P* (cos X i 2 ), 
1-0 
(106) 
0 21 m 0' 12 ” de d cos 6 0 = 4tt ± (- If {2k + l) ”*A* 21 "A* 12 P* (cos Xl2 ), 
1-0 
where the upper limit of k is the lesser of the two integers m, n. 
Applying these results to (89), (90), (92), (93), we have, therefore, 
(107) I (r.s,, Xl3 ) = W (2/im„)' +wl ' 
r+1, s+1 
•y J>+1A1 sAl i rAi S+1A1 
1- = 0 
— 2/x 2 C 3 R r A*i 2 i A* 12 (1 - cos xi 2 )} Pi (cos X 12 ), 
(108) i (r^, X12) = -yv(2/'<r s+i 
V + 1 , -S + 1 
2 (-1)* [f ,\ r+1 A* 21 °A .\ 2 +A ' A* a 1 A* 12 
1 = 0 
+ 2C 2 K r A / ‘ 21 s A / A {(miMo)''' ( 1 — cos X12) 
-i Pi (cos X12), • 
(109) J (rpq, Xu) = 17T 3 {2hm 0 ) r 
+ s+2 
, + 2 +J [’- +2 A ft 12 *A* 13 +t’- +1 A* J2 * +1 A* 12 + r A* 2 * +2 A A 1 , 
1 = 0 
-4/* 2 C 8 h ( r+1 AVA* 12 + r AV +1 A* 12 ) (1 - cos X 12 ) 
+ 4/* a 2 C 4 E r A* 12 *A* 12 (1 - cos X 12 ) 2 ] Pi (cos X 12 ), 
(110) J (7 Vl , Xu ) - fx 3 {2hm 0 ) r+s+2 
r+2, s+2 
■y 
i = 0 
2 (-1 Y [Mi2 r+3 A ft 21 * A a :2 +f r+1 A /i 2 p +1 A a ' 12 + / u 21 r A* 2 P +2 A* 2 
+ 2 C 2 k (fj-i2 l ' r+1 A /l 21 '’A* ] 2 +/u 21 / 2 'A*2i s+ 1 A /, 12 ) {2 (m ]^) 1 - (l —cosX12 
+ C 4 K r A A 2i s A A: 12 {2 (//1M2)' J (l —cos X12) — (miMs) -1 ' 2 } 2 ] Pi (cos xi 2 )- 
