228 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
9. There is another and simpler way of getting the factors of these 
continuants. For instance, if in T a we add to every odd column the sum 
of all the odd columns which follow it, and add to every even column the 
sum of all the even columns which follow it, then subtract from every row 
the second row above it, the determinant breaks up into (a 2 — /3 2 ) and a 
determinant of order (2 n — 2), which on interchanging the denominators of 
conjugate elements is a determinant of exactly the same form with n one 
less and /3 two more. Thus, if T a be represented by f 2n (a, /3), then 
/*»(«, /?) = (« 2 -/3 2 )/ 2 n- 2 («, £ + 2). 
In precisely a similar manner, if T & is represented by / 2n (a&, a/3), then 
f 2n (ab, a/3) = (ab - a/3)/ 2w _ 2 (a&, a + 2y . /? + 23), 
and if T c is represented by f 2n (ab, a/3, yS), then 
fojcib, a/5, yS) = {ab - a(/5 + 2 n- 2)8(y + 2n - 2 )}f 2n , 2 (ab, a/3 + 2/3, yS + 2y). 
10. It is easily seen that if T ft is represented by f n (a) . F^a), then 
fn( a ) = (a+2n- l) ./ n _!( - a) = (a - 2 n- l)(a - 2n - 3) ./„_ 2 ( a) 
and 
F n (a) = (a - 2n - 1) . F, t ( - a) = (a - 2 n- l)(a + 2n - 3) . F M _ 2 (a), 
which shows that the signs between the terms of the factors are alternately 
positive and negative for f n (a), and negative and positive for F n (a). 
The operations which show this are the following : — 
(1) Add all the other columns to the first. 
(2) Add to every column after the first the second column following it. 
(3) Subtract from each row the second row above it. 
(4) Subtract the first row from the second. 
(5) Interchange the denominators of conjugate elements in the reduced 
determinant. 
11. If in T & we put b — a, a = (3, and S = y, then it breaks up into factors 
which we may represent by f n (a, a, y) and F n (a, a, y), where f n {a, a, y) is 
the sum of two terms, and F n (a, a, y) the difference of the same two terms. 
Using the same set of operations, it is seen that 
/»(«> a > y) = (« + <*)/„_!(«, a + 2y, y) 
and 
F n (a, a, y) = (a- a)F n _j(a, a + 2y, y). 
That is, the linear factors of T & with the positive sign between the terms 
all belong to f n (a, a, y), and those with the negative sign all belong to 
F «(«> y). 
