1901 - 2 .] A Continuant Resolvable into Rational Factors. Ill 
tinuant the last horizontal difference is not /3 but J3 — b Q we have* 
x b 1 
b 2~ ft X b 2 
b 3 - b ± x b 3 
b 4 -/3 x b 4 
h~ b i x \ 
b 6 ~ft x 
x h 
b 2 ~P x h 
x \ 
\-ft x b 6 
b 5 -b 1 x 
h-p 
h 
x 
The two general theorems, one of which is illustrated by this 
example, are : — 
1. If in a continuant of the (2n)^ order the elements of the main 
diagonal be all alike , those of the upper minor diagonal be 
hi } b 2 , b 3 , ... , b 2n _i 
and those of the lower minor diagonal be so related to the latter 
that the horizontal differences are tq and /3 1 alternately save the 
last which is not /3 1 but /3 1 — b 2n , then the upper minor diagonal 
may be changed into 
h 3 , h 4 , h 5 , , h 2n _! , b 2n , tq 
without altering the value of the continuant. 
2. If in a continuant of the (2n + 1)* 71 order the elements of the 
main diagonal be all alike , those of the upper minor diagonal be 
bj ? b 2 , b 3 , • . . , b 2n 
and those of the lower minor diagonal be so related to the latter that 
the horizontal differences are tq and /3 1 alternately save the last 
which is not \ but tq — b 2n+1 , then the upper minor diagonal may 
be changed into 
bg , b 4 , bg , ... , b 2n , b 2n4 _j , /? 4 , 
without altering the value of the continuant. 
When in the one case b 2n = 0 , and in the other b 2n+1 = 0 , the 
theorems degenerate into that of the preceding paragraph. ^ 
(8) Another point of interest attaches to this transformation, 
however ; because it may happen that while the original continuant 
does not satisfy the conditions of § 3 requisite for resolvability into 
quadratic factors, the derived continuant may. 
* It should he noted that in the derived continuant the vertical differences 
are £ - b 2 , , /3 , b x , j8. 
