THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 319 
for particular types of molecular models (cf. § 10), appear not to converge at all 
rapidly. Fortunately, in our applications of the velocity-distribution function to the 
theory of viscosity and thermal conductivity, we need to know not the individual 
co oo 
values of the ft s and y’s, but only the sums 2 /3 r and 2 y r ; for these it is possible 
r = 0 r = 0 
to determine symmetrical expressions which are found, in practice, to be highly 
convergent. 
In what follows we shall use the symbol § mn placed before a function of the 
integral variables r, s (such as b rs or c rs ) to denote the (on, n) th successive difference 
of this function with respect to r and s respectively. Thus 
<U/(n s) = f(r, s)— m C 1 ( /(r— 1 , s) + m C 2 f(r—2, s)-... 
4, /(**, s) = f(r, s)- n Cif (r, s- 1) + „C 2 /(r, s-2)-... 
Lnf(r, s ) = <Lo f(r, s)- n G i S m0 f(r, s -1) + „C a $ m0 f (r, s- 2 ) -... 
= nf(r, s )-„A S 0n f (r—1, s) + m C a $ Un f(r-2, s)-... 
When we substitute b rs or c rs for f(r,s ) in the above formal expressions, any term 
with a negative suffix is to be omitted as being zero. 
Since the value of a determinant is unaltered by subtracting from the elements of 
any one row or column the corresponding elements of any other row or column, and 
since this process can be repeated indefinitely often, it is clear that from (141), by 
subtracting the (s — 1 ) th row from the s th , for all values of 5 from 1 onwards, we have*' 
(176) V (b n ) - V (S 01 b rs ), V (c rs ) = V (S 0l c n ). 
The same process applied to V r (b rs ), V r (c rs ) leads to determinants identical with 
V (S 01 b rs ) and V (^ 01 c rs ) respectively, save that in the r th column all the elements are 
zero except the one in the first row, which is unity. Evidently, therefore, V ; . (b rs ) 
and (c rs ) are the r th minors of determinants which are respectively identical with 
V (S 01 b rs ) and V (^ 01 c rs ), except that in each case all the elements of the first row have 
oo . oo 
the value unity. Consequently the sums 2 V r (b rs ) and 2 V r (c rs ) are equal to the 
r =0 r —0 
sums of the minors of the two determinants just described, i.e., they are equal to 
these determinants themselves. Thus, by (140), 
y o V (<^oi b r , ; ) y _ V (<bi c r.J 
J r V(J 01 U’ - = V(J M c r ,)’ 
where we have 
(178) b' 0s = 1, c' 0s = 1, (s = 0 to co), b' rs = b rs , c' rs = c rs , (s = Otoco, r = 1 tooo). 
VOL. CCXVI.—A. 
* When s = 0, 8 01 should be replaced by 8 00 . 
2 X 
