1887.] Mr A. H. Anglin on Summation of Alternants. 201 
of the form A+l, and where by equations (2) and (3) it will be 
found that 
S 2 = | 5, 4, 2, 1 | + |4, 4, 3, 1 |; 
and consequently 
'$[u+ 1, x+ 1, y, z] = | 5, 4, 2, 1 p . . . (5). 
Lastly, it may be shown in like manner that 
S|> + 1, x+1, y + 1, s] = (|5, 4, 3, 1I + |4, 4, 3, 2|)P 
= 15,4,3, 1|P (6). 
The results (4), (5), and (6) are the Extensions of equation (C), and 
their right-hand members are formed by deleting successively the 
second, third, and fourth figures from the series 5, 4, 3, 2, 1 ; that 
is, by deleting these columns successively from the corresponding 
matrix. We may further observe that if we increase by unity one 
index in the left-hand side of (C), the sum of the resulting alter- 
nants is obtained by increasing by unity the elements of one column 
in the right-hand side, thus giving the identity (4) ; while an in- 
crease in two and three indices produces respectively a corresponding 
increase in the elements of two and three columns, thus furnishing 
equations (5) and (6). 
3. To obtain the Extensions involving any number (v) of indices 
we should thus assume the corresponding results for v — 1 indices, 
and then deduce those for v. 
Now, in the case of n— 3 general indices r, s, t, . . . z, we have 
| a°b 1 c 2 d r e s . . . V | = | (r - 3), (s - 4), . . . , (z - n + 1 ) | p , 
which may be written in the form 
t, ... z\ = 
i 3, 4, 
5,... 
71=1 
ip, 
the Extensions of v 
diich are 
M 
+ 
h— ‘ 
• • z] = 1 2, 
4, 5, . 
. . n- 
■IIP 
. 
• (O' 
S[r+ 1, 8 + 1, 
t, ... z\ = 
1 2, 3, 
5,... 
71- 1 | 
£ ! • • 
• (2)' 
2[r + 1, 8+1, 
• • • 
t + 1 , u , . 
■■*] = 
1 2, 3, 
4, 6, . 
• 
. . n- 1 | 
P (3)' 
while generally 
S* = |2 
, 3, i, . . . 
/x+1, 
/x + 3, 
... 71 
- 1 IP 
• w 
and lastly, 
%+l, s+l,..+ 
+ 1, z] = \ 
2, 3, 4, 
. . . n 
- 3, n ■ 
- 1 1 £ ! • . . 
(n - 4)' 
