380 Proceedings of Royal Society of Edinburgh. [sess. 
This, however, if arranged in parts containing b° + c° , b 1 + c 1 , 
b 2 + c 2 , . . . . and their respective cofactors, is 
Wxn-I - 'R>c X n-fs + (b + C) { Xn _! - a Xn _ 2 + bc Xn _ z } 
- ( b 2 + c 2 ){ Xn _ 2 - a Xn _ 3 4- &‘ X n-J 
+ ( &3 + c 3 ){ Xn _ 3 - a Xw _ 4 + bc Xn _ 5 } 
Now it can be shown that 
a Xn-l - % bc Xn-2 = nK n , 
and, as we have already seen, 
Xn — 1 — ®Xn — 2 d" bc Xn-3 1 j 
we thus reach the following interesting result — 
/ b b 
The sum of the signed primary minors of l a a a ... a 
\ c c 
is the quotient of 
»K n + (6 + c)K n _ 1 -(^ + <ja) Kn _ 2 + (^^^)K f ^ lf - .... 
bya + b + c. (XVII) 
(11) It has recently been proved * that 
! a + d 
b + d 
d 
d ... 
& + d 
a + d 
b + d 
d ... 
d 
c + d 
a + d 
b + d . . . 
( - ) n+1 c n 
s(a - ]S) 
d 
d 
c + d 
a + d . . . 
n 
1 1 s + (n+\)d 
1 a n+1 1 + a + . . . + a n 
1 /3 n+1 l+/3 + ... + p n 
where s — a + b + c and a, /3 are the roots of the equation cx 2 -f ax + 
b = 0. Now, in the first place, the determinant on the left here is 
I b b 
by the third theorem (C) of § 1, the continuant (a a a . 
\ c c 
n 
increased by d times the sum of its signed primary minors : that 
is to say, is equal to 
K n + d- M(K n ) . 
In the second place, the determinant on the right is equal to 
1 a +1 
1 
1 
n+ 1 
+ d 
1 
a n+l 
1 + a + . . 
. +a" 
1 /3 w+ i 
1 
pn+l 
1+P + . . 
. +/S” 
* By Dr F. S. Macaulay in Math. Gazette , iii. pp. 44-45. 
