MONATOMIC GAS: DIFFUSION, VISCOSITY, ANI) THERMAL CONDUCTION. 151 
§ 9. First and Second Approximations in the General Case. 
(a) General Remarks. 
In the general case there are no such simplifications as have been described in §§7-9, 
and for practical numerical purposes we have to be content with successive approxi¬ 
mations to the complete solution. The approximate formulae become increasingly 
complicated with each successive step, however, so that it is fortunate that a first 
or second approximation generally gives a close approach to accuracy. In the theory 
of viscosity and conduction in a simple gas (‘Phil. Trans.,’ A, vol. 216, § 11, p. 334) 
it was found that a first approximation gave a result not more than 2 or 3 per cent, 
too small, whde the error after a second approximation was negligible. In the 
present theory of a composite gas the error of the first approximation may be much 
larger (up to 13 per cent.) in extreme cases where the masses, densities, or diameters 
of the two sets of molecules differ widely (cf. §7). Such a case is worked out to 
a fifth approximation in § 13 (e), and as this is one in which an exact solution 
is possible by another method, the results there obtained throw much light on the 
general character of the convergence of our analysis. It would seem that (as in the 
previous memoir just cited) the successive approximations form a monatomic sequence, 
the first and second members of which give a good indication of the accurate limiting 
value. If the difference between these two members is about 2 per cent., the 
additional correction due to all further approximations is about ^ per cent., while 
if the difference is so much as 8 per cent., the further correction is about 4 per cent., 
the additional correction being in an increasing ratio to the first difference as the 
value of the latter rises. 
In this paper we shall not go beyond a first approximation in the general case, 
except in regard to a' 0 , where we shall stop at the second approximation. 
(b) First Approximation to a' 0 . 
If in our set of equations (5"08) we neglect all save the central one, and consider 
only the central term of that, we get the equation 
(9 Ol) a m (a 0 a _o) — I) 01 C£oi) a 0 — kAs 1 
By (5'12) and (4‘27) we have 
_ 1 2Trv l i/ 2 m 1 m 2 KUO] 
x 27v 0 (m 1 +m 2 ) 
It is convenient to choose A 0 (which is quite arbitrary as yet) so as to make 
^00 = 
(9‘02) 
(9’021) 
so that 
(9-022) 
27 (m 1 + m 2 ) 
27rA 1 A 2 i' l j7n 1 m 2 K / 12 (0) 
