1901 - 2 .] A Continuant Resolvable into Rational Factors. 109 
where it will be seen (1) that the elements of the main diagonal of 
the continuant are each equal to x ; (2) that the elements in the 
odd-numbered places of the upper minor diagonal and those injthe 
even-numbered places of the lower minor diagonal are 
b, l-d, b-2d, b-3d, ... 
-d, -2d, -3d, ... 
respectively ; (3) that, when counting in the opposite direction, the 
elements in the odd-numbered places of the lower minor diagonal 
and those in the even-numbered places of the upper minor 
diagonal are 
0, 0-8, 0-28, 0- 38, . . . 
-3, -23, -33, ... 
with their signs changed ; and (4) that the factors are 
(x 2 + b0) ( x 2 + b^~d- 0^8) (x 2 + b^2d- 0^28) 
(4) What the corresponding theorem is in the case where the 
continuant is of odd order will readily be seen from the example 
x 3b 
- (3 x c-v2d 
-b x 2b 
— (3 — d x c + d 
-2b x b 
— [3 -2d x c 
-3b x 
= {x 2 + 3 b(0 + c + 2d)} . {x 2 + 2 b(0 + c + d)}-{x 2 + b{0 + c)} • x . 
(5) When in §3 0 = b — half the order-number of the continuant,, 
and 3 = d = 1 , we obtain a continuant having factors resembling 
those of Professor Elliott’s continuant, viz. 
x p 
- 1 X p-l 
- 1 X p-l 
- 2 x p - 2 
-p + 2 x 2 
-p + 1 x 
-P + 
= (x*+l 2 )(x 2 + 2 2 )...(x 2 +p 2 ), 
1 
X 1 
p X 
