MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 139 
In order to make this investigation complete, from a purely mathematical 
standpoint, it would clearly be necessary to supplement the above formal solution by 
a discussion of the questions of convergence raised in the course of our analysis. The 
complexity of the problem, however, and the rudimentary condition of the theory of 
infinite determinants, may well absolve the author from an attempt at such a task, 
for the present. From the standpoint of mathematical physics there is, fortunately, 
sufficient numerical evidence (cf. § 13e, f and ‘ Phil. Trans.,’ A, vol. 216, p. 330, 
Table III.) to afford reasonable assurance that our expressions converge satisfactorily ; 
this is especially so in regard to the formulae actually used in the applications of 
/(U, V. W), ie., (5'22)-(.V25). 
(c) On Certain Combinations of the Coefficients a, (3, y. 
For the purpose of the theory of diffusion, viscosity, and thermal conduction in 
composite gases, we require only certain combinations of a, (3, y, and never their 
individual values. The following expressions comprise all those we shall find 
necessary in this paper; in connection with them we may refer back to (3T3), (3T4), 
and the formulae (3'03), (3‘04) for f( U, V, W):— 
(5-22) 
* 
(5-23) 
(5-24) 
a'o = “^2 (a r —a_ r ), 
o 
fi'o = — A X A 2 j (A)—/3_o) + 2r _1 (/3 r —/3_ r ) 
2,h f 1 m 1 3 Uy — O x 2 + v^mf> U 2 C' 3 2 ) — iLQa 2 ( v iyr3~ 
(5’25) 2h + v 2 m 2 \3 2 G 2 2 ) = -f f 1 a 0 + v 2 a_ () ) + 2(r + f) (v 1 a r +v 2 a_ r ) 
-IB. U {j („ A + ^s_ 0 ) .+ £ r±i ( n fs r + «£-,) 
A,5'., 2 r (i' l a, + «*_,) + B, Us (»i/3 r + »2^_,) 
We therefore desire to obtain concise expressions for the following quantities :— 
(5'26) i(* r -*_ r ), 
0 1 
(5‘27) 2r (v 1 a. r + v 2 a._ r ), 2 (* ffi r + v 2 /3_ r ), 2 (^y r + v 2 y_ r ). 
1 1 0 
The denominators in the expressions (5*21) for a r , (3 r , y T are independent of r. Hence 
our problem consists in the combination of the numerator determinants V r for an 
infinite number of values of r. 
VOL. OCXVII.-A. 
X 
