304 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
Applying this result to (91), identifying A, B, C with one or more of the points 
c 0 , c R , c' R , and P with x, we may with but little difficulty prove that 
//fill U?Uf d cos 0 R d<p 0 d<p R = {0i 2 2 + 40i 2 0' 12 + 0' :2 2 — 4// 2 C K 2 (0 12 + 0' 12 ) (l — cos X 12 ) 
+ 4// 2 2 C k 4 (l - cos X12) 2 }• 
Similarly we may show that 
Mi 
U? d cos 0 1£ d<p {> d<p R — f 7r 2 0' 
12 ) 
so that 
Mi 
UV (3Uf — Of) d cos 0 R d<j) 0 d<p R — {0i2 2 + §0120^2+ 0 r i2 2 
— 4// 2 C 1; 2 (0 12 + 0 7 12 ) (l — cos X12) + 4m 2 2 C r 4 (l — cos Xis) 2 }- 
Hence we have 
(92) J (r.Sy, X 12 ) = 57r 2 (2hm 0 ) 
r + s + 2 
012 2 + f 0120^2 + 0\ 2 ' — 4// 2 C r 2 (0 12 + 0' 12 ) ( 1 — COS X 12 ) 
+ 4//.fC R 4 (l — cos X 12 ) 2 } 0'i2 S 0i2 r de d cos 0 O , 
and it may be proved in a similar manner that 
(93) J (?Vl 5 X 12 ) — f ( 7T-'(2 hm 0 ) r + S+2 | | [mi2021 2 +§0210*12+ M210 / 12 2 
+ 2C r 2 (Ml 2 1/2 02 1 + M21 V, 0 / 1 2) {2 (miM2)' /2 (1— COS X 12 ) — (miM2) _V: } 
+ C R 4 {2 (m!M 2 ) 1/2 (l— cos X 12 ) — (miM 2 )- 1/2 } 2 ] 0 / i2 S 02i r de d cos 0 O . 
The Expansion of (p 2 + E—2pcr cos 6) n in a Legendre’s Series. 
§ 7 (D) In order to effect the integration of I and J with respect to e and 0 O we must 
have recourse to the expansion of 
(94) P„(/l, °") COS 0) EE (/f + ar“ — 2/1(7 COS 0) n 
in a series of Legendre’s functions. In a recent paper* I have shown that 
(95) P„(p, 0-, COS 0) = 2 (- l) 7 ' (2^+1) n A . k ( p 2 , rr 2 ) P*(cOS 0 ), 
k =0 
where P /f (cos 0) is the ordinary Legendre’s function of cos 0, of type Jc, andf 
(96) 
”A*(m 2 , - 2 ) = 
(p Y y (m+ §)«_* 2 (r—i) 
V £l(i+i), («-*),., p 
jf j ]}t (11 + -> )' a (j— t ) 
,pj t=l: (t + §) t (t — k) t _ k 
a 2t 
* Chapman, ‘Quarterly Journal of Mathematics,’p. 16, 1916. The expansion is there not limited to 
integral values of n, though these are alone considered in the present paper, 
f The constant k is necessarily a positive integer; if k > n, n A k = 0. 
