320 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
To the above determinants we now apply the same process of differencing by columns 
which has already been applied by rows, and we thus obtain the equations 
(179) 
y R _ V (Snb'„) 
v(j u 6„)' 
CO 
r = 0 
v (Jn <4) 
The determinants V (S n b' rs ) and V (S n b rs ) are identical save in their first rows; all the 
elements of the first row of the former are zero save the first, which is unity. Hence 
V (f,,//,„) is equal to the principal minor of V (S n b rs ) ; we shall denote it by 
V' (S n b rs ). Hence, and with a similar notation for the principal minor of V (S u c rs ), 
we have # 
GO 
■V 
r = 0 
Pr = 
Z (W 
v (Snb„) ’ 
_ V' (SuC„) 
V (''ll'’,.) 
All these determinants have now regained a symmetrical form. 
It is convenient, partly for the sake of elegance, and also because it imparts a 
highly convergent form to the elements of our determinants (cf. § 10) to continue 
this process of differencing still further, as follows. We repeat the whole of the 
above operation of differencing by rows and columns an indefinite number of times, 
beginning now at the second row and column (thus leaving unchanged the values 
both of V and its principal minor), and afterwards successively at the next later row 
and column than on the previous occasion. The general element thus becomes S rs b rs 
or S rs c rs , and we have 
(180) 
y' 
V (SJJ ’ 
Z Ra.) 
where the dash (') denotes the principal minor of the corresponding determinant. 
These expressions could, of course, have been obtained directly by a re-arrangement 
of the original equations of transfer, but it seems preferable to use the latter in the 
more simple, natural forms chosen, and to make this transformation by differencing 
in relation to the determinants formed by the elements b rs , c rs . 
§ 9. Consideration of Particular Molecular Models. 
§ 9 (A) While, as we have seen, certain general properties of the elements b rs , c rs can 
be demonstrated without the assumption of any property of the molecules save 
spherical symmetry, it is possible to carry our investigations much further when we 
represent the molecules by particular models of simple type, such as point centres of 
force, or rigid elastic spheres. This involves, primarily, the examination of the 
functions 0 2/c (ry). 
* When r or s is zero, the corresponding suffix of 8 n should also be written as zero. 
