860 Proceedings of Royal Society of Edinburgh. [sess. 
represented by B'. In the upper right-hand corner we have p + k 
rows of elements represented by B, followed by r -Jc rows of 
elements represented by X. The elements in the lower left-hand 
corner are the same as their conjugates. 
Our theorem applied to the determinant thus formed would be : 
(2p + q\p + r\h\q \/2p + q\p + r\r + k\k\ 
\ a i Si ft A a ] a 2 i ' 
2p + q \p + r\f2p + q\p + r\r - Jc\f2p + q\p + r\r - 
a, a 2 / \ a 2 i) 
ff) 
(p+q-r-h. 
\P+K-h / 
i 
where P* denotes the product of 
/2 p + q\p- s rr\r-k 
V a, a, 
1 2 
( c lp + q\p + r h \ 
/2p + q\p + r\h\p + K-h\ 
V “i 111) 
\ a i Si 'i ) 
and the aggregate 
(2p + q\p + r \h\q\ k' 
\ a i Si S 2 P 
(2p + q\p + r \r- k\i 
) 
'2 p + q \p + r\h\p + Jc-h\ 
{2p + q\p + r\h\q\n\ 
\ <h a 2 )\ 
v a i Si i ) 
V <*i fiiPtj) 
O) 
X 
3=1 
If h = 0 the relation becomes 
cr) 
Z ( - 1 )’ ■ 
/p+q-r\ 
\ p+q ) 
= Z(- 1 ) >! 
(i) 
Z(- x )" 
3=1 
a n a 0 
2 p + q p + r r - K\{2p + q\p + r\r - k\k 
/2p + q\p + r\ q\(2p + q\p + r\r - k\k 
\ a i ft A 
(2p + q\p + r^ 
/2 p + q\p + r\r- k\ 
V a i a 2 ) 
2p + q p + r \p + k 
( X-i O-o i 
(2p + q\p + r \q\ k 
\ a i ft J 
2p + q\p + r\r- *\(2p + q \p + r \p + + q \p + r \ q | k 
«i ft J 
