482 
Proceedings of Boyal Soeiety of Edinhurgli. 
we see that n must be 5, and that the said determinants are all 
derived from 
‘^1-2 
<^1-3 
«l-4 
^1*5 
'2*1 
«2-2 
eo 
«2-4 
^2-5 
^3-1 
%2 
%3 
«3-4 
%-5 
^4-1 
®4-2 
%4 
^4-5 
«5*2 
^5-3 
^5*4 
S-5 
The details of the process of derivation are recalled in connection 
with the interpretation of the pairs of suffixes. A requisite pre- 
liminary is to form all the different pairs of the numbers 1, 2, 3, 
4, 5 ; arrange them in the order 
12, 13, 14, 15, 23, 24, 25, 34, 35 , 45 ; 
and then number them 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
These last are the numbers from which the suffixes are taken, and 
what each one as a suffix refers to, is the combination under which 
it is here placed. For example, the first suffix in refers to the 
combination 1 2, and implies the deletion of all the rows of the 
above determinant of the fifth order except the 1st and 2nd j the 
second suffix refers to the same combination, and implies the dele- 
tion of all the columns except the 1st and 2nd; and the symbol as 
a whole thus comes to stand for 
^ 1.1 %*2 
Cto* 
1 ^ 2-2 
(XLI. 2.) 
Interpreting \ , .... in the same way, we see that 
(e) 
is a compact notation for the system of which the determinant is 
1^1* 1^2*21 ? 

. . . , 
l‘^l- 4 %* 5 l 
[a3.4a4.5l 
l%*l%*2l > 
, jag. 3^5.4! , 
1 ^ 3 * 3 ^ 5 * 5 1 ’ 
[a3.4a5.5j 
1 ^ 4 - 1^5* 2 ! 5 
i% 3 ^ 5 ' 5 l > 
[a4.4a5.5j 
