1904-5.] Continuants whose Main Diagonal is Univarial. 509 
6<f> + — 
w 
+ J 2+ & 3 0^ + b t + b b - 
M e+i i + h 
6 + _^ 3 _ 
4> + 
= d * +b i+h 
1 + 
Consideration of the case where n is odd leads to the same result, 
— a result given, probably for the first time, by Heilermnan.* 
(Ill) 
n 
= {6(f) - 2 2 ) (0<f> - 4 2 ) {6(f) - 6 2 ) when n is even, 
and == 6{6<h - l 2 ) (6cf> - 3 2 ) {6<f> — 5 2 ) when n is odd, 
—a theorem which degenerates into Sylvester’s {Nouv. Ann. de 
Math., xiii. p. 305) when <£ is put equal to 6. It has to be noted, 
however, that the mode of proof followed in the case of Sylvester’s 
theorem, viz., removing the linear factors separately, is now unsuit- 
able. A mode of removing the quadratic factors will be found in 
the Proc. Roy. Soc. Edin ., xxiv. pp. 105-112. 
(5) Thirdly, if we denote the preceding generalisation of 
Sylvester’s continuant by cr n we obtain 
6 1 .. 
x <f> 2 . 
. x-1 6 3 
. x - 2 <f> 
* See Zeitschrift f. Math. u. Phys., v. (1860), pp. 362-363 ; also Giinther’s 
Darstellung der Ndherungswerthe der KettenbriLchen, p. 75, Leipzig, 1873. 
n{n - 1) 
(x n + 1 )cr n _ 2 
(I 
+ 
n(n - 1 ){n - 2 ){n - 3) 
“ 2*4 
{x - n + 1) (x - n + 2)e 
+ 
n{n- 1) ... (n- 5) 
— (x-n+ 1) {x - n+ 2) 
{x-n + 3)o 
(4) Similarly we have the theorem 
6 1 .. 
n- 1 </> 2 
n - 2 6 3 
n - 3 <f> 
