112 Proceedings of Royal Society of Edinburgh. [sess. 
In illustration of this we may take a continuant which has 
points of resemblance with Professor Elliott’s, viz. 
x ip 
- 1 x ip - 1 
- 2 x ip -2 
- 2p + 3 x 2p + 3 
-2p + 2 x 2p + 2 
-2^5+1 
in which the horizontal differences are all 4 p except in the last row 
where the difference is 4 p - (2p + 1) , and for which therefore we 
may substitute 
x ip - 2 
- 1 x ip -?> 
- 2 x ip - 4 
- 2p + 3 aj 2p + 1 
-2p> + 2 x ip 
-2p + l x 
But this, if the last column be divided by 2 and the last row 
multiplied by 2, is a continuant with the constant horizontal 
difference ip - 2. It therefore follows that 
x 2 + (ip - 2) 2 
is a factor of it ; and as the co-factor is a continuant of the same form 
as the original, the process may be continued with the final result 
{x 2 + (ip - 2) 2 }{x 2 -\- (ip - 6) 2 } . . . {x 2 + 6 2 }{x 2 + 2 2 } . 
The analogous result for the case of an odd-ordered con- 
tinuant is 
x ip + 2 
— 1 X ip + 1 
- 2 x ip 
- 2 p + 2 x 2p + 2 
- 2p + 1 x 2p + 3 
-2 \p x 
— {x 2 + (ip) 2 } [x 2 + (ip - i) 2 } .... {x 2 + i 2 }x. 
{Issued separately April 1 , 1902 .) 
