THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 305 
In the last equation the symbol^, where q is integral, is defined thus :— 
(97) Pg = p(p-i) (i>-2)... (_p-g + i). 
From (53) it is clear that 
(98) {0 12 (cos 0)Y l = P„ ( Ml C 0 2 , mA/,- cos 0 ) = £ (2&+l) "A* 12 P, (cos 6 ), 
k =0 
(99) {0 21 (cos 0)}” = P„ (m 2 C 0 2 , mA 2 , cos 6) = A ( — l)* (2£ +1) H A* 21 P /c (cos 6), 
where we have written, for brevity, 
(100) ”A ; h 2 = a A /c ( Ml C 0 2 , M2 C R 2 ) 
"A* 21 = “A* ( M2 C 0 2 , m A 2 )- 
In our expressions for AQi, 0 takes the values A and 0' o , and the variable e 
is involved only through the latter angle, which occurs in 0 r ]2 or 0 12 (cosA o ). 
In the expansion of the latter (cf. 98) in terms of P ; . (cos A 0 ), or, by (48), of 
P A . (cos @ 0 cos X ]2 + S in 0 O sin X 12 c °s c), we shall make use of the following well-known 
formula in the theory of spherical harmonic functions :—• 
(101) P 7 . (cos 0'„) = P A . (cos A cos xi 2 + sin A sin x‘ 12 cos e ) 
= P A (cos A) Pa (cos X 12 ) + 2 A ( cos 6 ) ( cos X 12 ) cos le. 
The Integration with respect to e and 0 O . 
§ 7 (E) Since the integral of cos le with respect to e, between 0 and 2 tt , is zero unless 
1=0, from (98) and (101) we deduce the result, 
(102) [-0V de= 2tt £ {2k + l) "A* 12 P a (cos A) ?a (cos Xl2 )- 
Jo l :=0 
Now from (89), (90), and (92), (93), it is evident that as far as concerns integration 
with respect to e and A we have to consider a number of terms such as 
(103) j j 0 m 0' 12 ” de d cos A, 
where 0” 1 may have the suffix 12 or 21, while 0'” always has the suffix 12. Now 0™ 
does not involve e, so that (102) suffices for the integration with respect to e, and 
leaves us with 
(104) 2ttJ ^ jz (±1)*(2£+1 )“A*P*(cos A)} 
