THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 315 
I have little doubt that, with rather more trouble, c rs could be shown to share the 
above properties of b rs , but I have not made any serious attempt to prove this ; from 
the numerical calculations in § 10 (A) it appears probable that the increase of c rs with 
r, s is more rapid than that of b rs . 
Properties of the First Row or Column of V (b rs ) and V (c rs ). 
§ 8 (D) The numerical values of b rs and c rs obtained in § 10 suggest that many 
further general properties of these elements might be determined, with sufficient 
trouble, and that the convergence of the determinants V ( b rs ) and V ( c rs ) might thus 
be,demonstrated. Owing to the considerable algebraic difficulties involved, however, 
I have so far made little progress towards the proof of such properties, except for 
the case when r or s is zero, i.e., for the elements of the first row or column of 
V ( b rs ) and V ( c rs ). It will be shown that 
(152) b r0 = b 0r ~ o r0 = c 0r 
for all values of r. 
This will be proved as a particular case of the more general result that 
(152a) (s + l) b rs (k) = c rs (k ) when the lesser of r and s is even, and k = [r, s], 
where b rs ( k ), c rs (k) denote the parts of b rs and c rs respectively which are due to a 
particular value of k in (143), (144), while [r, s] denotes the upper limit of k, as usual, 
i.e., k — fe+l or ^-s + 1, whichever is the less. Thus if we suppose that r = s, and 
that s is even, (152 a) takes the form 
(153) (s+l)6 rs (|-s+l) = c rs (|-s + l). 
When s = 0, this value of k is unity, and b r0 ( l), c r0 (l), which usually form only 
a part of b rs , c rs , become the whole, so that (152) is the particular case of (153) 
corresponding to this value of s. 
Since B /l (r, s ) is zero when either r or s is less than k, some of the terms in 
b rs (i’S-t l) 5 c rs(l' 5 +l) vanish. In fact, as may readily be seen from (143), (144), we 
have 
(154) Mi5+1) = 32B 0 !A. rs || e~ (x ‘ +y " ) F +2 (ry) |b s+2 (r+1, s + 2) 
+ " ^ + ^ r B s+1 (r + 1 , s +1) \ x\f dx dy, 
Zs “T o 
B 5+2 (r, s + 2) + 4 ( g + 2) y*B s+1 (r, s + l) 
2s + 5 
+ 
4 (s+1) (s + 2) 
(2s + 3)(2s + 5) 
x 2 y 2 dxdy 
(155) c„(js+i) = ne,v\' r . 
