THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 303 
The Integration with respect to 0 R , <j> R , 0 O . 
§7 (B) In I (rs, xi 2 ) and J (rs, xi 2 ) it is clear from (48) to (53) that <j> R does not 
appear at all in the integrand, while 6 1{ and <p 0 occur only in the products U\U and 
U'i 2 (3U L —C-) respectively. We have 
U^lf = C 0 2 cos 2 X + /x 2] C R 2 cos 0 R cos 0' R -t- ^-CoCk cos X (cos 0 R + cos 0' R ), 
U'iUo = C 0 2 COS 2 X — C H 2 COS 0 R COS 0' R + C o C R COS X (w>i* COS d f u — /jL V 2 ,J cos 0 E ), 
and, remembering the values of cos X and cos 0 f R , we have 
j U iUi cl cos 0 R cl<p 0 d(p R — | n 2 {C 0 ‘+/x 2 iC R “ cos X 12 —m 2 i , 2 C 0 C R (cos @ 0 +cos 6f 0 )}, 
||| U\U 2 d cos 6 r d(p 0 d<p R = f-7r 2 {C 0 2 —C E 2 cos Xi 2 + QAi (^ 21 1/2 cos 0 'q—M i 2 Vi cos 0„)}. 
In the notation of (53) the latter two equations may be conveniently re-written as 
follows :— 
Mi 
U', lh d cos e R d(p 0 d<p R = IiT (0 12 (cos 0 O ) + 0 12 (cos 6' 0 ) - 2/x 2 C R 2 (1 — cos xi 2 )}- 
' r r 
(miM 2 ) v ' 2 11 U'iU, d cos 0 R dfa d < p R = fx 2 {mi 2 ’ /2 0 21 (cos 0 o ) + m 2 i V '0i 2 (cos 0 ' o ) 
+ 2 (miM 2 ) 1/ 'Gr 2 (1 — cos x) — (miM 2 )~ ,/2 C r 2 }. 
Substituting in (81) and (82), we thus have 
(89) I (r x s 1} xu) = fx 2 (2 hm 0 ) r+s+1 ff {0 12 +0' 12 -2 M2 C R 2 (l - cos Xl2 )} 0'u ©i 2 r ded cos 0 O , 
(90) I (r 2 s u X 12 ) = i-r 2 (2 hm 0 ) r+s+1 j {Mi 2 1/i 02i + M 2 i I/2 0 , u + 2 (miM 2 ) 1,2 C r 2 (l — cos xi 2 ) 
— (miM 2 ) _,/ “C r 2 } 0 / i 2 ®0i 2 r de cl cos 6 0 
§ 7 (C) In the case of J (rs, xi 2 ), we have 
(91) , Ml 2 U?Ui 2 = (mi 1/2 C 0 cos X + « 2 1/2 C r cos 0 r ) 2 (ni /2 C 0 cos X—m 2 i/2 C r cos 0' r ) 2 , 
in which (cf. the figure on p. 293) X = c 0 O;c, 6 R = c R Ox, 0' R = c' R Ox. In the integration 
over the sphere, with respect to 0 R and <p 0 , since 6 0 , e, xi 2 are constant the triangle 
c 0 c R c' R preserves its form, so that we may, if we please, regard x as the variable 
point and cp R c' R as fixed. Now it may readily be proved, by the method of 
“ poles ” in the theory of harmonic functions, that if A, B, C are three fixed points 
on a unit sphere, and P a variable point, then the integral over the spherical 
surface of 
cos 2 PA cos PB cos PC 
is 
xg-Tr (2 cos AB cos AC+ cos BC). 
2 T 
VOL. CCXVI.-A. 
