354 DR THOMAS MUIR ON 
be the diagonal adjacent to the diagonal of units in the set,—that is to say, of the 
set be 
Oo Oseh OF oO 
—and vf we denote the elements of the resolvable continuant by their place-names (1,1), — 
(1,2), ... , then the factors of the continuant are 
ul, 1) +&- (1, 2)} 
{(2, 2)-& -(1, 2) +& - (2, 3)} 
-{(3, 3)-&- (2,3) +&-(3, 4)} 
{(n, 2) —&,-1:(n-1,n)}. . a ae 
A scrutiny of the procedure connected with the removal of any factor makes this 
evident. For, firstly, when s—1 factors have been removed, the residual determinant 
has for its first column the line of column-multipliers last used, viz. 
secondly, this determinant when reduced to the next lowest order has 
(s,s) — €1-(s—1, 8) 
for the first element of its diagonal; thirdly, the employment of the next line of 
column-multipliers, viz. | 
changes the said element into 
(s,s)-& 1: (s-1,s)+&(s,s+1); 
and this, in virtue of the character of the process, is the next factor ready for removal. 
It may be noted in corroboration of § 6 that the sum of the factors thus expressed is — 
(1,1)+(2,2)4+@,3)4+ ...+4(m,n). 
(16) Observing from the foregoing that 
(n,n) —&,.(n—1, 2) 
is the last factor, we have suggested to ourselves the obtaining of the factors in the 
reverse order by the use of a set of row-multipliers, the first operation being 
HON = Se OOM et ae ; 
An interesting result is thus reached, viz., that corresponding to each set of column-— 
multipliers for the resolution of a continuant there is an equally effective set of row- 
multipliers. 
Thus returning to the continuant of § 2 and performing the operation 
row, — 8row, + 28 row, — 56 row, + 35 row, 
we find we can remove the factor a — 8b — 8c, and write its cofactor in the form 
a 2°4D : : : | 
5(b+c) a-2e 3(b — ¢) 
6(b+2c) a-8e 2(b-2c) 
; 7(b+3c) a-18e 1(b-3c) 
35 — 56 28 -8 Lene 
