346 DR THOMAS MUIR ON 
(6) The sum of the elements of the main diagonal of either of the continuants m 
S$ 2, 3 as equal to the sum of the factors into which the continuant is resolved. . (V) — 
This is true of any continuant of the form 
atk, py 
Yq Gt, po 
Qo G+, 
that is resolvable into factors linear in a. By way of proof we have only to note (1) 
that since the diagonal term is the only term of the continuant that contains either the 
n'™ or (n—1)™ powers of a, it follows that the coefficient of a”~' in the continuant 
IS @+%,+%,+ ..., or 2x say: and (2) that if a+mu,,a+m,,a+m,,... be the 
factors into which the continuant is resolved, the coefficient of a”~* in their product is 
My + Mothg+ ...,0r Desay. We thus have 
Le = Zp, 
and .". nma+ Le =na+ Dy, 
as was to be proved. 
(7) The full table of multipliers used in § 2 is found to be 
ied: Teale ene eee 5 
Tingle uOalG hen eee Ont 
ley 6: W202 tee meee pe Cha: 
a 
ite 8, oe ow We » gz Ors2,5 
Il, BS fi DF Cras7 
—in other words, each multiplier is of the form 
C4515 28-19 
and the question next arises whether the continuant resolved in § 2 is the only one 
which this set of multipliers is capable of dealing with. In order to make suitable 
answer we have to ascertain the relations which must exist between the twelve 
quantities 
B, Bo, Bs, By 
Pi 0 Pe aes 
, ; 1A PE SE Ie 
in the continuant 
a 2-48, 
5y, atp 3P, 
6y, atg 28, - 
: Tage ict 8, 
: 8y, ats 
in order that it may be resolvable into linear factors by means of the operations of § 2. 
