THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 301 
it is easy to see that (18) is satisfied by this form of F, while in order to satisfy (17) 
we must have 
(75) /?_!+ 2/3 r /(r+1) = 0. 
0 
The products B„/3 r , B„y r are quite definite, but B„ and C 0 can evidently be assigned 
arbitrarily; we shall decide that their values, though unspecified for the present, are 
alike for F l and F 2 . 
The above expression for F (U, V, W) is equivalent to that obtained by Enskog (§ 1), 
by an entirely different method. But the chief difficulty of our problem, and one 
hitherto unsolved, lies in the determination of the coefficients /3 and y ; this is effected 
in the present paper by means of AQ. 
§ 7. The Calculation of AQ,. 
§7 (A) In calculating AQi we shall deal chiefly with A 12 Qi (cf. § 5 (A) and (67)). 
The particular forms of Qj which we shall consider are 
(76) Q: = (2/im 1 ) s+J U 1 C 1 2s = B/ s> , 
(77) Qr = (2/im 1 ) s+1 U 1 2 C J 2s EE 
In accordance with §5 (D), the only part of F(U, V, W) which is relevant to 
A3Si (s) is 
(78) 
_ B 1 ^ ui- 
1 CX o 1 
(2 hm) r 
3.5... (2r + 3) f 
&-1C 
2r 
while that which alone concerns ACi U) is 
(79) 
— 2hmC 0 (c n U 2 + c 22 V 2 + c 33 W 2 ) 
(2 hm) r 
1.3.5... (2r+ 5) 
y 
.0 
2 r 
As to the latter, since the remainder of the integrand of ACi (s) is symmetrical with 
respect to V and W, the parts of this integral arising from V 2 and W 2 in (79) are 
equal, so that c 22 V 2 + c 33 ,W 2 can be replaced by ^ (c 2 2 + c 33 ) (V 2 + W 2 ) = — |-c n (C 2 — U 2 ), 
by (74). Hence for our purpose (79) is equivalent to 
(80) 
■|(2k)C 0 Cn(3U 2 —C 2 ) 
(2 hm) r 
1.3.5... (2r+ 5) 
Vr 
C 2r . 
We shall denote by b 12 (riSi) the part of A 12 33 i ( s) which arises from the term 
— (2hmi) r+i UiCi 2r in Fi(Uj, Vi, Wi), and by 6 12 (r 2 Si) the part arising from the corre¬ 
sponding term of F 2 (U 2 , V 2 , W 2 ), in each case the numerical and other factors in F 
