200 Proceedings of Royal Society of Edinburgh. [april 18 , 
which, in the same manner as before, may he shown to he equal to 
4 3 1 
3 3 2 
4 3 1 
+ 
3 3 2 
4 3 1 
3 3 2 
and thus 
S[3+l, y+1, z] = \4r, 3, 1|{* . . . (3). 
These two results are the Extensions of equation (B), and are 
formed by deleting successively the second and third figures from 
the series 4, 3, 2, 1 ; that is to say, the second and third columns 
from the matrix 
x — ?H- 4 , x-n + 3 , x — + 2 , x-n - (- 1 
y -n + 4 , y — n + 3 , y - n + 2 , ?/ - % + 1 
z-^ + 4, z — w + 3 , 2 - w + 2 , z - % + 1 . 
2. Again, in the case of four general indices u, x, y, z, writing the 
equation 
| cdb l c 2 . . . f n ~ 5 g u h x kH z | = | (u — n + 4), (x — n + 3), y - n + 2), (z — n 4- 1 ) | if 
in the form 
[u,x,y,z\ = |4, 3, 2, lj£-, .... (C), 
we have 
%[u + 1, x, y , = 
2 consisting of four terms in each of which there is one index of 
the form X+ 1, and where 
5 4 3 2 
4 3 2 1 
4 3 2 1 
4 3 2 1 
4 3 2 1 
+ 
5 4 3 2 
+ 
4 3 2 1 
+ 
4 3 2 1 
4 3 2 1 
4 3 2 1 
5 4 3 2 
4 3 2 1 
4 3 2 1 
4 3 2 1 
4 3 2 1 
5 4 3 2 
Expanding these determinants as before, it may be shown by the 
application of equation (2) that 
S, = | 5, 3, 2, 1I + |4, 4, 2, 1 | ; 
and we thus have 
S|> + 1, x, y, z\ = | 5, 3, 2, 1 | £* . . . (4). 
Farther, we have 
%\_u + 1 , x + 1 , y, z\ — S 2 £* suppose, 
where % consists of six terms in each of which there are two indices 
