of Edinburgh, Session 1876 - 77 . 
383 
• *• * * 
• • * * ■* 
#■ * *■ 
where the dots and asterisks denote zero elements and non-zero 
elements respectively. 
In the solution of this, determinants of a set of other forms 
require to be considered, viz., — 
• 
* 
* 
% 
* 
• 
* 
* 
* 
* 
• 
* 
* 
* 
* 
• 
• 
* 
* 
* 
*• 
• 
* 
* 
% 
• 
• 
* 
* 
• 
• 
* 
* 
* 
• 
• 
* 
* 
* 
* 
• 
* 
*• 
* 
* 
• 
• 
* 
* 
% 
• 
• 
* 
* 
* 
• 
• 
* 
* 
* 
* 
• 
• 
* 
% 
* 
• 
• 
5 
* 
* 
* 
• 
• 
in all of which the elements of the main diagonal are zero ; the 
elements of the adjacent minor diagonal being in the first also all 
zero, in the second all zero except the first element, in the third 
all zero except the second element, and so on. Let us denote the 
number of terms in these when they are of the nth order by 
XoO) , XiO) , xM , 
and let the number of terms sought be 
Further, let any determinant-form with dots and asterisks stand 
for the number of terms in such a determinant; the four forms 
above, for example, being thus symbols equivalent to ^(5) , Xo(5) , 
Xi(5) , Xa(5) , respectively. 
3 E 
VOL, IX. 
