140 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
(d) Difference-Transformations of Infinite Determinants. 
It is convenient at this stage to describe certain operations, by the application of 
which we are enabled to preserve the symmetry and increase the convenience of our 
determinantal formulae. These operations will be termed “ continued differencing ” 
by rows, by columns, or by rows and columns, and we shall denote the corresponding 
symbolic operators by S 0t , ^ r0 , or S rs respectively. We shall first define them in relation 
to infinite determinants which cover only a quadrant of the infinite plane (rfi> 0, sf> 0), 
and afterwards in relation to the more complicated type which occurs in this paper. 
The operation of continued differencing by rows (§ 0s ) or by columns (<fi 0 ), applied 
to the determinant V (f rs ), where rfi> 0, sfi> 0, transforms it into the determinant 
v 0,,/j or V (o„j,:) respectively, where 
(5-28) S,J„ = i (-1)\C,<W„ = 2 
n = 0 m — 0 
We may effect the operation S 0s as follows : from each element of row s we subtract 
the corresponding element of row s — 1, for every row from 5=1 onwards : this done, 
we repeat the operation on the transformed determinant, except that we now begin at 
s — 2 : and this process is continued without end, beginning each time with the row 
next after the initial row on the previous occasion. It may readily be seen that the 
result is as we have already stated, and that the value of the determinant is 
unaffected by the operation. Continued differencing by columns is strictly analogous, 
and need not be separately described. Continued differencing by rows and columns 
is performed by applying the two separate operations successively, the order being 
immaterial. Without alteration in value, the determinant is thus changed in form 
from V (/„) to V (<?„/„), where (cf. (5'28)) 
(5'29) S r .fr. = i*,ttX) = (WJ = 2 2 (-!)”« 
m = 0 n = 0 
In the case of determinants which are infinite in both directions, the operation of 
continued differencing by rows and columns is effected by applying the process 
described above^to each separate quadrant; thus the differencing by rows is performed 
by differencing outwards from the centre row in both directions (above and below), 
and likewise, by columns, both to right and left of the centre column. Neither of 
these partial operations, nor the complete process, alters the value of the determinant, 
a fact which we may express by the equation 
(5-30) v (/„) = V (J rt /„) = V (4/„) = V 
where the notation is similar to that used in the former case. There, however, r, s, 
and consequently also m, n, were necessarily positive or zero, while our convenient 
notation for determinants of the present type involves also negative values of r and s. 
