378 Proceedings of Royal Society of Edinburgh. 
D 
character. Towards finding this we first note the following 
property of continuants, viz. 
The cofactors of the elements in the places 
(n, n) , (n - 1 , n ) , (n— 2 , n ) 
of the continuant 
( h \ 
. (i , ») 
or 
are 
K n - 1 > k «-2 ’ * 
. , /q , 1. 
Changing K n into the form ( ^i c i ~ b 2 c 2 
& & n yq a 2 a 3 . 
and putting in the said places of it 
1, K 15 K 2i . . . . , K n _, 
we thus learn that the resulting determinant is equal to 
(K«_ lf K n _ 2 , . . . , IP, K 15 . . . , K n _ 1 ) 
and therefore is equal to x«-r other words we have 
Xw-l 
(XI) 
a 
-be 
• • K n _ x 
- 1 
a 
-be 
. . k w _ 2 
-1 
a 
. . K n _g 
a K x 
. 
. - 1 1 
(XII) 
This determinant, however, may he developed in another way, viz., 
in terms of the elements of the first row and their respective co- 
factors ; and doing this we obtain 
Xn — 1 ^Xn — 2 &%-3 1 (XIII) 
— a recurrence-formula which readily gives 
^_ i=zna n-\ _ (?j, _ l)C n _ 2 , 1 a n ~ B bc + ( n-2)C n _ B , 2 a n ~ 5 b 2 c 2 - ... (XIY) 
In illustration let us take the case where % = 4. We then have 
/ b b b \ 
the sum of the signed primary minors of I a a a a I 
\ c c c / 
= Xs “ ( b + c )x2 + ( ft2 + c2 )xi - (b 3 + c B ), 
= 4 a 3 - 6abc — (b + c)(3a 2 — 2bc) + (b 2 + c 2 )2a - (& 3 + c 3 ), 
= 4a 3 — 3 a 2 (b + c) + 2 a(b 2 - 3 be + c 2 ) - (b + c)(b 2 - 3bc + c 2 ) . 
