660 Proceedings of Royal Society of Edinburgh. [sess. 
The rest of the paper (pp. 13-24) is occupied mainly with con- 
tinued fractions, and contains nothing new in regard to the related 
determinants. Possibly the nearest approach to a fresh result is 
the use of the known identity 
1 + 
%~ a i + „ + 
to suggest (p. 16) that 
& 0 - «i - 1 
a x a z a 2 - a s 
= a x a 2 . . . a n _j j a Q -a 1 +a 2 
. i 
Gunther, S. (1873). 
[Darstellung der Naherungswerthe von Kettenbriichen in inde- 
pendenten Form, iv + 128 pp. Erlangen.] 
When we come to Gunther we have reached the first formal 
treatise on our subject. His booklet consists of three chapters, 
the second of which, with the title “ Darstellung der Zahler und 
Henner jedes Naherungsbruches in Determinanten-Form,” is that 
which mainly concerns us. The first chapter (pp. 1-30) is of the 
nature of a historical review of the various modes proposed for the 
calculation of convergents, and the third is professedly an applica- 
tion of the results of the second chapter to “ Analysis, Algebra, 
und Physik.” 
That portion (§ 6) of chapter i. which deals with the introduc- 
tion of determinants into the treatment of continued fractions 
attributes the first idea of the existence of the relation to Ramus 
(1855) ; notes that Heine (1859) discovered it independently; and 
nevertheless represents Heine as having used a result of Painvin’s. 
All this, of course, stood at the time * in need of serious modifica- 
tion, which unfortunately was not forthcoming, with the result 
* For Heine’s repudiation see his Handbuch der Kug elf unction, i. (1878), 
pp. 261-262. 
