352 DR THOMAS MUIR ON 
Further, there is a similar difference in the expressions for ¢: here the expression is _ 
— 98, + 256, - 14f,, 
whereas in § 8 when given in terms of 8,, 83, 8, it is 
~ 98, +258, - 148,.. 
The second theorem is—The continuant 
i ite), : 7 
(n+2)y, A, (n—2)8, fae 
(n+ 3)y5 A (73) Ce are 
(n+ 4)y5 Nl ante Sa ea ee 
as resolvable into linear factors if 
B,=b-(m—lj)e + a a) Td, 
2(2m + 1) 
(m+ 3)(m + 2) 
Ym= b+ (m+2e = cca Ne 
me : ; 
A, =@—2(m?=1)c + ae = 5(2m? + 2+5n)-7Td, 
the s factor being 
a+ 2(n —2s+1)b-2(s—1)(2n - 284+ 3)e 
+(n-—s+4)(s—1)-7d, 
or 
{a + 2(m—1)b} — 2(s — 1){2b — 5c} —(m=s+4)(s—1){4e-7d}.  . . ie 
An immediate deduction from this is that when 14d=2b + 38¢ the s” factor is 
a+2(n—1)b+4(m—s)(s — 1)(20 — 5c) ' - 
and vs the same as the s” factor from the end, so that when in addition n is even the 
continuant is a square. ; ; (XII) 
The third theorem is—The value i any continuant of He form referred to in § 3 
is not altered by adding to its matrix the matrix of the continuant 
= ine i (n= Ve : ; io oer aj 
~ Flute -F(n-Te E(u de if 
~ Ferd -2(n-1e F(n-Be .... - <I 
= F (n+ 4e - (n= 31l)e 
ik 8 ena Ge er ee ee 
horns a hE ee AG ‘ 
Sh : Claes ie aera Gels ; 
NDT ag ee Gene 
(Oi RP oe. Gee 
will be found in the Trans. S. Afr. Philos. Soc. referred to above. 
