356 DR THOMAS MUIR ON 
example, in the case of the 5" order the equivalent tables of column-multipliers and — 
row-multipliers are 
[eee [> Sa epee 
ie eG ame eet eee 
jee, 6 ike at ’ 
ir, Lape 
1 
(18) There falls now to be noted a set of theorems regarding resolvable continuants — 
of a totally different form but connected with and derivable from those of §§ 2, 3, 13. 
If in any one of these latter theorems we put 0 for the element in the place (1, 1), 
the continuant is expressible as the negative product of the elements in the places — 
(1, 2), (2, 1), and a continuant of the lower order n—2: further, one of the said — 
elements is contained in the first factor of the original continuant and the other in the — 
last factor : in this way, therefore, the resolution of the new continuant of order n—2 is — 
secured. ‘Thus, taking the five-line continuant dealt with in § 2 and putting a=0 we © 
obtain . 
-24.50(b-c) | 8e  2(b+ 2c) 
7(b — 3c) 18¢ 1-(6 + 3c) 
; B(b—4e)  32e 
= 80(4b + 14c)(20c)( — 4b + 18c)( — 8b + 8c) , 
and therefore 
8¢ 2(b + 2c) 
T(b—3c)  18e “ 1-(b+3e) 
a = (40 + 14e) - 20 - (— 40 + 180). 
B(b-4e) 32¢ 
The general theorems thus obtained are 
A (GG Says ae Ree ieee ae 2(n—1 1b = 2(2n+ e+ 14n-+2)-5a | 
(n+ 4)y, Ay (fh A Ey ar | {2(n — 3) )b- 4(2n —1)e+ 2-(n + 1)-5d} (XIX) 
{2 a 
(1+ 5)y2 era e Rithcohs < | (n — 4)b — 6(2n — 3)e+ 3-n-5d} 
n ~ 
: Ese _, mm + 1) ee 5), _ (m+ 2)(m+ 3) 
if Bp=b-(m+l1je+ 22m + Bia > Y¥n=O+ (m+ 2)e 2eOnaam +3) dd, 
x ‘ 9, 2 2m(m+2)+64+3n, | 
Ae 2. —% ee : 
and n=(m+1) De a 
s RST og eae n+4)74 | 
A, (n-1)8, ik ee = 4.05 zs (nm — 1) Bn + Bet Lat 4)-Td 
(n+ 6)y A, (= 2) BO wa. ew {2(m — 3)b — 4(2n + 1)e + 2-(n + 3)-7d} (XX) 
(n+7)y. Ye Diep Fk {2(n — 5)b — 6(2n — 1)e + 3-(m + 2)-7a} 
: Le oy _(m+4)(m+t 5) i, 
Le B,, =! (m+ bec ORE es Ym = b+ (m+ 4)e— ~ 2(2m+5) 7 
2 
and An =(m+1)(m+ 3) J | —2%e+ aan omasy d } ; 
