CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 353 
When the first theorem there given is attempted to be generalised in the manner 
employed in the present paper the following is the result :— 
The continuant of the n” order whose main diagonal rs 
a, at+(b—c)—(n—2)y, a@+2(b-c)—2(n-4)y, a+3(b-c)-3(n—-b)y,.... 
and whose minor diagonals are 
(n-1)b, (n-2)(b+y), (m—3)(b+2y),....- 
@s 2(c-y), (eat) 
ws equal to the product of the n factors 
{a+(n—1)bd} 
-{at+(n—-2)b-c+1-(n—2)y} 
-{at+(n--3)b—-2¢+2-(n—-3)y} 
os Chae ; pe : am 
It is seen to degenerate into the original theorem when y¥ is put equal to 0. If, how- 
ever, we write X in it for the half-sum of the first and last factors, and Y for b+c, 
the factors may be written 
X+43(m-1)Y, 
X+h(n—3)V+1-(n—2)y, 
X+4(n—-5)Y4+2-(n-3)y, 
Ra 
thus showing that four variables are not necessary for the expression of the identity. 
An easier way of reaching the same result is to put 
a=a—(n—1)é, 
b=B+é, 
c=B-€, 
when it will be found that € appears in the continuant but not in its factors; and 
when there are consequently obtained at one and the same time the case of theorem 
(XIV) where b =c, and an expression for the corresponding nil-factor continuant. 
(15) We have thus in all at present three sets of column-multipliers, each of which 
has associated with it a linearly resolvable continuant of the form 
a By | 
Nm @t+p BP, 
y2 atq_ Bs 
Other sets will doubtless be discovered, as the only difficulty is the devising of a set 
which will not lead to unreasonably complicated expressions for the elements of the 
—continuant. In all cases of 
ons & &3 > hd gay ie Fine Gen 
