552 Proceedings of Royal Society of Edinburgh. [sbss. 
the factor a-b , a 1 being left in its place. Our theorem is thus 
established. 
(9) The determinant derivable from G(a , a , ... , b ,b) in the 
manner just stated may he denoted temporarily by G (g , v). It is 
formed most expeditiously by writing the /x units in each of /x + v 
rows as if forming a circulant, and then completing the first row 
with v units : thus 
G(2,3) = 
11111 
. 11 .. 
. . 1 1 . 
. 1 1 
1 . . . 1 , 
It is a determinant with very peculiar properties, and the 
evaluation of it is a matter of considerable interest. 
(10) In the first place we have to note that /x being > 1 
G(/x,/x) = 0. (L) 
This is evident from the fact that if to the second row we add 
the ( g+ 2) th , we obtain a row of units exactly like the first row. 
In the second place there is the theorem 
G(l,/x) = l. (2) 
Here we have a diagonal of units and zeros in every place on one 
side of it. 
In the next place there is the more interesting result 
G([x , v ) = G(v , f) (3) 
which is not so readily evident. To show the equivalence of the 
two determinants take either, subtract its first row from each of its 
other rows, then change the signs of the latter rows, and finally 
remove the first row to the last place. Thus the first operation 
performed on G(2 , 3) of the preceding paragraph gives 
1 
1 
1 
1 
1 
-1 
-1 
-1 
-1 
- 1 
-1 
- 1 
_ 1 
-1 
-1 
-1 
-1 
• 
