THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 311 
The term corresponding to k = 0 is absent in both the above cases, since <j>° n (tuV) 
is itself zero, so that 9 '/ u (ni y) is the function (j> n (r n y) of lowest order ( k = 2 ) in 
b n (nsi) or Cn (nsi). The upper limit of k in the case of b n (n^i) is equal to the 
integral part of the lesser of the two quantities \ {r+ 1 ) and ^ (s+ 1 ); this is denoted 
by <V, s ). Similarly the upper limit of k in the case of c n (nsi) is the integral part 
of the lesser of the two quantities |-(r + 2 ) and |-(s + 2 ), which we denote by [r, 5]. 
Thus, when r = 0 or s = 0 , b n (r x s 1 ) = 0 . 
We can now write down the complete expressions for AQ, in the two cases above, 
as follows:— 
(132) 
(2 hm,i) s+1 
1.3.5... (2s + 3)s 
AUA 
2s 
1 0T 00 l 
= rp "5 2 , 1 {&11 &12 (^l^l) } +Ml2^/^r-l,2^12 (^2^1)]? 
1 OX r = 0 
(133) 
(2hnii) s+1 
1.3,5... (2.S + 5) 
AU^Cj 25 = C 0 c n 2 X'„ [y rtl {c u (nSi) +c 12 (nsi)} + y,, 2 c 12 (>Vh)]. 
r = 0 
In the present paper we are concerned with the application of these formulae only 
to simple gases, in which v 2 = 0 and hence b 12 (r^) = b 12 (r 2 s 1 ) = c 12 (r 1 s 1 ) = c 12 (r 2 Si) = 0 . 
It is convenient to write the reduced equations in the following form :— 
(134) 
(135) 
_ (2 hm) s+ “ _— A l jC 2(s+1) — (LL 2 B b 
1.3.5... ( 2.5 + 5) (s+ l) , T dxrtf r r 
(2 hm) s+1 45 
1.3.5... (2.s + 5) 2v 
^ AU 2 C 2s = Cn 2 y r C r 
r = 0 
In (134) we have substituted r+ 1 , s +1 for r, s in (132), multiplied by 3/v, and 
used the notation given by 
(136) b rs EE — B 0 X r s 6n [r +1, s+1); 
the first term in (132), with factor /3_i, vanishes, since b n {0,s) = 0. 
(135) we have written 
(137) 
QA rs C U ( ? ’l 5 l)‘ 
Similarly in 
§ 8. The Expressions for the Coefficients in the Velocity- 
Distribution Functions. 
§ 8 (A) We have now obtained expressions for AQ, the rate of change of a function 
of the molecular velocities due to encounters, in two different ways : in § 3 AQ was 
found from the equation of transfer, while in §§ 4-7 it has been determined by direct 
calculation. By comparison of (26) and (134)—substituting s +1 for s in the former 
—and of (30), (72) and (135), we deduce from these different expressions for AQ the 
YOL. CCXYI.-A. 2 U 
