MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 147 
(7 06 ) P 2 \{ r 2 S z) — 4 7r,/ l ,/ 2/ u l2 r + ' (r + S+-§ ) r + , + ]^ r + ,K / i 2 (o), 
(7 07) p i 2 {?\$\) = ir, 7ri/ i I/ 2 M 2 ^ r+1 (rs + r-{-s + %) (r + s + §) r+ , + iK / 12 (0), 
(7'08) 
(7’09) 
(MO) 
Piiir&i) = T 5 7 r,/ i l ' 2 /Wi 2 r+, rs(r + -§) r+2 (s + 2 ').,^ r+ iK / 1 2 (o), r> 0 , s> 0 . 
t 
P 12 
/ 
P 21 
/ 
P 21 
(0 2 Sl) — (s+y),.^ (~^o + T^^ 0 ) (0) — p 2 \ {s x 0 2 ), 
( r 20l) = 4't' 7ri, l l/ 2/ u 22 r ("T + f ) r + 2 ( —^ r + T 3 5^ r 12 ) K'i2 (0) = p 2X (0iV 2 ), 
(n« 2 ) = T5 7 n/ i l/ 2Mi2 r+ Vs (r+|-) r (s+-|), + 2^ j+ iK / 1 2 ( 0 ), r> 0 , s> 0 , 
(*Va) = ^x«/ii/2At 1 2 r+, (r + s + f)r + , +8 *, 3 r+I+1 K , 1J (0). 
For the particular values of r and 5 there dealt with, the expressions (4‘26)-(4'34) 
reduce to the above in the present special case, as may readily be verified. 
We now substitute from (7‘03)-(7'10) in (5'12)-(5'20), retaining only the terms of 
the highest order, and neglecting quantities of order m 2 : m x ; the result is as 
follows :— 
(7 11) S^>>0, d 0s — -2^7rAi\ 2 i / o'm/iM2-^-ol£ 12 ( 0 ) — C*00) S<^0, Ct 0l — '^7' 7r '^A2 l 'u^ , i/ A 2-^-o^'»K- 12 ( 0 ), 
(7' 12 ) r > 0, s>0, a„ = — N r ,p :1 (tvs,), 
^0 
(7-13) 
(714) 
(7-15) 
(716) 
(717) 
(718) 
(719) 
T ^ 6, S 0, Ct TI 2^, TrAjA^l'Q'TTi,^, A,;, / , " b* & r + ,K. 12 (0), 
W + yMs + yL 
{? ^ 0, s 0), ci r , = i^fT\i\ 2 v () 'm 2 p. l A. (1 k r K. 12 ( 0 ), 
(r > 0, s < 0), a„ = -^ T 7r\ ] \ 2 v Q m 1 p. 2 A 0 k s K , l2 ( 0), 
(r > 0), 6 r = 0 , & = -^l^iW^ifeBo ^M- 1 K/ 13 (0), (r<- 0 ). 
(»> 0), 6, =0, 6. = -L*^\ I V„m ]J » J B 0 ^tirlK' 1! (0), (*<-0), 
A 2 -tv 5 1 X 
(r<-0, 3>0), &„ = 0, (r>0, s<-0), 
(r> 0, s> 0), 
b rs — A 1 A 2 ^0 W/ 2MlB| 
A! it 
(r+l) (s + l) 
(r + s+-|) 
/r + s 
(r + %)r(s+%)i 
fo r +s + 2 ^r + 1 ^* +1 
+1 K' 1S (0). 
Since, by (714), (7’ 15), a rs (r < 0, 3 >• 0) is independent of 3 , and a„(r>0, s < 0) 
is independent of r, it is readily evident that in the alternate quadrants of the 
determinants V (<^ mn a mn ) and V' ( S mn a mn ) all the elements vanish. Hence each V becomes 
VOL. CCXVII.-A. 
Y 
