350 DR THOMAS MUIR ON 
is also 
X1X = ew 
{KAY 2 een 
. {X — 6Y - 3-(n — 2)Z} 
X= (On =) Ne neem ez! 
where 
X=a+2(n-1)b, Y=2b-3¢, Z=4c-5d. 
And as a, 6, c, d cannot conversely be expressed in terms of X, Y, Z alone, the 
left-hand member of the identity, that is, the continuant, can only be made to appear 
as a function of X, Y, Z and one of the four a, b, c, d. Consequently, supposing 
this to be done, and thereafter all terms involving X, Y, Z deleted, we shall obtain 
a continuant which not only vanishes but which can be viewed as having n vanishing 
factors. 
(11) To obtain this nil-factor continuant there is, however, a better method. For, 
as it is the special case where X , Y, Z vanish, it must be obtainable by putting 
4e = 5d, 
2b = a 
a= —I(n-1jb= —-(m—- Ned, 
or, therefore, by putting 
8 
@ = =e, 
15 
2 
eS =8, 
3) 
(h = @, 
Doing this we find from § 9 
Oe ary ale ae 
‘ees 2{n+2(m—1)?-1} 
(2m — 3)(2m — 1) 
and have the following theorem :—The value of any continuant of the form spoken 
of in § 9 ws not altered by adding to its matrix the matrix of the continuant 
Ge ee We 0 
— ne 2A (n+ l)e n—2) 56 3) Sree fee F 
—(n+ Nee a(n + 7)e (n - 3)h¢ dhe ee s (VIN) 
9 
n+ 26 ee er 
