1904-5.] Dr Muir on the Theory of Continuants. 
661 
that some of the inaccuracies continue to he propagated in text- 
books to the present day. 
In the second and third chapters neither pains nor space is 
spared to furnish an exposition of known results, with careful 
citation of the titles of all writings made use of, whether important 
or unimportant.* Throughout the two chapters, however, only 
one new and noteworthy property in regard to continued-fraction 
determinants is brought to light, the source of it being an identity 
attributed to Heilermann,f viz. 
x + i t)n , = x + b A - 
4 --^ bo 
Vv 
^ 3^4 
* + T+ 
*+h+h- x ± li+bb _ 
where the number of b’s appearing on both sides is n. Clearly the 
numerator of each convergent on the left must have an intimate 
relation with the numerator of one of the convergents on the right, 
and this Gunther doubtless sought to discover. The result which 
he established is 
X 
1 
-1 
b 2 
-1 
X 
h ... 
-1 
i .... 
as.-' b. 2n _ 
- 1 1 
x + b 1 - \b 2 
— 1 X + &2 + ^3 — ^ 3^4 
. — 1 x + b 4 + b 5 
where the order-number of the determinant on the right is half 
that of the determinant on the left. 
The demonstration occupies as many as five pages (pp. 70-74) ; 
probably the character of it will be sufficiently understood from 
seeing it applied without comment to the case where n = 2. The 
steps are — 
* The total number of footnote references in the book is 116. 
t Zeitschrift f. Math. u. Phys., v. (1860), pp. 362-363. 
