1901 - 2 .] A Continuant Resolvable into Rational Factors. 107 
which, clearly, is equal to 
x 5 
-1 x 4 
.-2x3 
. - 3 x 2 
- 4 x 1 
. -5 x . 
But as the second factor here is exactly of the same form as the 
determinant with which we commenced, we are thus led at once 
to the final result 
x 7 
— 7 x 
X 
7 
1 X 
5 
X 
3 1 
I X 
1 
-7 
X 
1-5 
X 
-3 
X 
1 -1 
X I 
(x 2 + 7 2 ) • (x 2 + 5 2 ) • (x 2 + 3 2 ) • (x 2 + l 2 ). 
In the case of an odd-ordered determinant the penultimate 
result is 
x 6 | x 4 
x 2 
— 6 x 1 — 4 x 
- 1 x 1 
.-2 x 
which finally gives 
( x 2 + 6 2 ) ( x 2 + 4 2 ) ( x 2 + 2 2 ) x . 
(3) Now with this process as a guide, let us deal with the much 
more general determinant 
x b x 
“ft " X ^2 
-ft x h 
- /? 5 x b x 
“ft * h 
~ ft x h 
- /? 2 x b 7 
— ft X 
Multiplying row-wise and then column-wise as before we obtain a 
result which is seen to he resolvable into a continuant of the 2nd 
order and a continuant of the 6th, provided we can satisfy the 
equations 
