1903 - 4 .] Dr Muir on the Theory of Continuants. 
131 
(the mode of formation of which is self-apparent), these 
successive coaxal determinants will be 
A 1 1 
J 
1 
0 
> 
A, 
1 
0 
0 
1 a 2 
1 
■A-2 
1 
1 
A2 
1 
0 
0 
1 
^■3 
0 
1 
A3 
1 
0 
0 
1 
A 4 
1 , Aj , A]A 2 — 1 , AjAgAg — Aj — A 3 , 
AiA 2 A 3 A 4 — A 4 A 2 — A 4 A 4 — A 3 A 4 + 1 , 
AiA 2 A 3 A 4 A 5 — A 4 A 2 A 5 — A 4 A 4 A 5 — A g A 4 A 5 — A 1 A 2 A 3 
+ A 5 + A 3 + A x . 
It is proper to introduce the unit because it is, in fact, the 
value of a determinant of zero places, as I have observed 
elsewhere.” 
After using this as an aid to prove his proposition regarding 
Sturm’s theorem, he returns to his new determinant in the 
following words : — 
“ I may conclude with noticing that the determinative 
[determinantal ?] form of exhibiting the successive con- 
vergents to an improper continued fraction affords an 
instantaneous demonstration of the equation which connects 
any two consecutive such convergents as 
and viz. 
J-h— i Lb 
For if we construct the matrix which for greater simplicity 
I limit to five lines and columns, 
A 
1 
0 
0 
0 
1 
B 
1 
0 
0 
0 
1 
c 
1 
0 
0 
0 
1 
D 
1 
0 
0 
0 
1 
E 
and represent umbrally as 
a i 
h 
a 4 
h 
h ; 
