1913-14.] 
Some Factorable Continuants. 
223 
XVI. — Some Factorable Continuants. By W. H. Metzler, Ph.D. 
(MS. received May 15, 1914. Read June 15, 1914.) 
1. In the Transactions of the South African Philosophical Society for 
January 1905, Dr Thomas Muir gives the most general continuant resolvable 
into factors by means of a given set of line-multipliers. He starts with 
the multipliers and determines the continuant resolvable by them. At the 
end of his paper he gives another continuant and its factors, but not the 
line-multipliers, which he says “ is equally interesting in itself and equally 
full of promise as a base for investigation.” Throughout his paper Muir 
is dealing with one of the two determinant factors of order n into 
which every centro-symmetric continuant of order 2 n can be broken up. 
Starting with the larger continuant of which Muir’s is a factor, one of the 
objects of this paper is to determine a set of row and column multipliers 
that will cause the continuant to break up into quadratic factors and 
thence into linear factors. Other and more general types of con- 
tinuants are given which these same multipliers reduce to quadratic 
factors. Another and more convenient way to determine the factors 
of these determinants is obtained in the form of reduction formulas. 
It is also shown how, for the two parts of order n into which the 
larger continuant of order 2n breaks up, the linear factors come out 
by reduction. 
2. The determinant in question is 
T = 
a 
(~n- l)fi 
2rc-l 
!.(/? + 'In - 2) (2rc-2)(/?+l) 
3-2 n 'In - 3 
■ 2.Q8+2rc-3) (2rc-3)Q8 + 2) 
5 - 'In ! 'In - 5 
(2»-2)(/i+l) „ l.(/3 + 2»-2) 
2/4 — 3 3-2 n 
(2m- l)j8 
2n- 1 
a 
2 n 
{a 2 -£ 2 }{a 2 -(/? + 2) 2 } . . . (a 2 -(/3 + 2rc-2) 2 }, 
