MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 131 
of a, /?, y determined in the succeeding sections will, however, hold good also in the 
general case ; the determination of the S ’s will be found in the Appendix. All the 
corrections to (f ) 0 are of the first order, and hence are separately deducible ; 
the complete value is obtained by adding to the series in a, /3, y, calculated in the 
body of the paper, the series in <5, determined in the Appendix .—June 2, 1916 .] 
§ 4. Completion of the Equations of Transfer. 
(a) The Values of AQ. 
We may now complete the ecpiations of transfer (2'07)-(2T0) by the insertion of 
the values of AQ, calculated in terms of the constants in the expressions for y(U, V, W) 
given in § 3. The calculation of AQ is a lengthy and elaborate operation which will 
not be described here, since a full account of it is to be found in my second memoir 
(‘ Phil. Trans.,’ A, vol. 216, § 7, p. 301). It appears that only those terms in f which 
are of odd degree in U, V, W contribute to the resulting expression for AUC 2 *, and 
only the even terms, similarly, contribute to AU 2 C 2s . The following results will be 
quoted forthwith :— 
(4-01) miAUA* = 2 
2 ,+1 (s + f) s+ i‘ 
r = 0 L 
3T \ 
my (N r A£Vr + sN' rs B 0 — f3 r J {p n (nsi)+/>i 2 (nsi)} 
/ 0T \ 
+ m 2 1 N rs A 0 ^ / 0 a_ r + sN' rs B 0 — /3_ r ) p l2 (? 2 s i) 
(4-02) 
mjAUjC, 2 * = 2 
r = 0 
0T \ 
m, (N„ A 0 £ Vr + sN' rj B 0 — f3 r j p 21 (ns 2 ) 
+m 2 (N rs A 0 £ 0 a_ r + sN rs B 0 ^-- {p 22 (f 2 s 2 ) + p 2 \ Qv> 2 )} 
(4‘03) ilu'TlV f AU AV = i( V„ 2 N"„[y r {/, 1 (»' lSl ) + / u (r lSl )} + r V,(w)], 
2 V 5 + f/*+a2vi V\ 
r = 0 
(4-04) U AUa » c > = Ic o c« 2 W\,[ y y n (r A ) + y_AMrA)+MrA)n 
2 t®+f;*+a2i'a v 2 
r = 0 
( h) Explanation of the Notation. 
In the above equations r and s may take all positive integral values, including 
zero. We shall presently alter our notation so as to consider also negative integral 
values, but in the following definitions of N rs , N'„, N" rs , the positive numerical 
values of r and s are in every case to be used on the right-hand side : 
(4'05) N r , = {2 r+s+2 (i 
+ f)r 
hi ( S + l)s+l} 
-1 
N' = — N 
x.-* rs - 1 - 1 rs ? 
rs 
N" = N 
r+l, s + l* 
In the case of N'„, the factor r is to be omitted from the denominator when r = 0 ; 
when 8 — 0, the value of sN' r „ in which form occurs in (4‘0l), (4'02), is to be taken 
VOL. CCXVII.—A. 
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