MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 129 
of v and h (but not their derivatives) as factors. We suppose them to be capable 
of expansion in power series as follows* (see Note C, p. 196):— 
(3-05) ^((V) = £ « ■ , (V, F 2 (C 2 2 ) = £ a_ r — C 2 S 
r = 0 1 ' 
3.5...(2r+3) 
r = 0 
1.3.5...(2r+3) 
(3-06) Gj(Cd) = £' -- 0 ^ C, 2 b G 2 (C 2 2 ) = 2' /3_ r 
(2 hm 2 ) r 
r = 0 
I .3.5... (2r + 3) r 
r = 0 
.3.5... (2r + 3) 
r 
p 2r 
^2 5 
(3-07) H,(C, 3 ) = £ y r -■ 0 C > Jr . 0,(0/) = i y_ r - 
r = 0 
1.3.5... (2r + 5) 
r = 0 
1.3.5... (2r+ 5) 
p 2r 
• 
/o-a>ti\ t /p 2\ _ x' ? ( 2hmi) r p 2r t /p 2\ _ v' ? (2 hm 2 ) r p 2 
(3 071) J 1 (G)- f 2^ ligi 5 i i(2f+l) G . J2 (° 2 >— J 0 ^ 1 .3.5...(2r+1)° 3 
The dash (') after the sign of summation in (3'06) is used to signify that the 
factor r in the denominator of the numerical coefficient is to be omitted in the first 
term (r = 0). The choice of the notation +r and — r for the suffixes has a convenience 
which will become apparent later; we may remark, in passing, that for this purpose 
a distinction must be maintained between + 0 and — 0. 
In (3'03) and (3'04) the constants A 0 , B 0 , C 0 , D 0 can be chosen arbitrarily, but 
when this has been done, the remaining constants a, /3, y, $ all become perfectly 
definite. 
( b) Relations between the Coefficients. 
The velocity-distribution functions ./(U, V, W) must satisfy the three conditions 
expressed by the equations (with appropriate suffixes 1 or 2 throughout):— 
(3‘08) 
(3'09) 
jj | /(U, V, W) d\J dM d\N = 1, 
/(U, V, W) C 2 d(J d\ d\N 
2 hm 
(hi or h 2 ), 
(3-10) l/ULh, Vi, WOlhdlhdVidWi = Ui-u 0 = u'Jx u 
jj|/ 2 ( u 2 , V 2 , W 2 ) V 2 d\J 2 dV 2 d\N 2 = u 2 -u 0 = -u'J\ 2 
* [Here, and throughout the remainder of the paper, where h and T appear without any suffix, they are 
to he read as h 0 and T 0 .-—June 2, 1916.] 
