654 
Proceedings of Royal Society of Edinburgh. [sess. 
and P r is n times the complementary minor of the element in the 
place (1,1) of Q r . In the latter form it is a relation between 
continued-fraction determinants, and as such claims our attention. 
By way of proof Bauer increases each row by all the rows 
following it, and thereafter diminishes each column except the 
last by the columns immediately preceding it, and thus obtains 
— n 
1 
a r + 1 
— n — a l 
a l 
1 
OL r -f- 1 
-n- a 2 
a 2 
a r + 1 
a r _! 
a r + 1 
— —n — a r _ x 
a,. 
and therefore 
= — TzDj + (n + cq) • | D 2 — (a r + 1) * II | , 
if Dj be put for the determinant got by deleting the first row and 
first column of Q r , D 2 for the determinant got by deleting the first 
two rows and first two columns, and II for ( n + a 2 ) (w + a 8 ) . . . 
(n + a r-1 ). Similarly, of course, 
P r = n -n 1 . . a,. + 1 
— n — a 2 a 2 1 . a r + 1 
a-r-i a r + 1 
. . . — n — o. r _ i ci r j 
and thus by altering the (1,1) element into a 1 — (n + cq) there is 
obtained 
P r = - n(n + a 1 )J) 2 ; 
so that on the elimination first of D x and then of D 2 from these 
expressions for Q r and P r there results 
P r +Q,-(» + a 1 )| (l-»)D s -(<v+l)Il| 
P,. + «Q r = n | (1 - (n + aj) (a r + 1)11 i 
These, on returning from , D 2 , II to their lengthy equivalents, 
are changeable into 
