of Ediiiburgli, Session 1883-84. 
389 
2. The Eesearches of M. E. de Jonqui^res on Periodic 
Continued Fractions. By Thomas Muir, M.A. 
1. During the present year there has appeared at intervals, in the 
Comptes Rendus of the French Academy, quite a series of com- 
munications by M. E. de Jonquim’es, on the subject of those periodic 
continued fractions which are the equivalents of the square roots 
of integers. These communications have attracted attention, both 
on account of the number of results given in them, and because, as 
a writer in the Bulletin des Sciences Matliematiques says, of their 
interesting and profound character. To any one really intimate 
with the bibliography of the subject, this cannot but be a little sur- 
prising. It is true that the number of so-called theorems is great ; 
but the very special character of a number of them, the fact that 
they are just such theorems as may be obtained by experiment and 
induction, and the want of demonstrations of them as evidence that 
the author was in possession of a mathematical theory of the subject, 
are points that have been too much overlooked. Further, and what 
is more important, many of the theorems are not new, and there is 
a sense in which the epithet “ new ” cannot fairly be applied to any 
of the earlier ones, because of the existence of a widely general 
theorem in which they are directly included, or from which they 
may with readiness be deduced, 
2. It is to this general theorem I now wish to direct attention, 
making use of it for the purpose indicated, viz., of giving scientific 
order and unity to M. de Jonquieres’ work. The theorem iras given 
in the year 1873, in the first paper I had the honour of communi- 
cating to this Society, and is to be found at p. 234 of vol, viii. of the 
Proceedings. It, as well as several other theorems given in the 
paper, was originally accompanied by a good deal of detail of the 
same nature as M. de Jonquikns’ theorems ; these special proposi- 
tions, however, were struck out, when by request an abstract of the 
paper was prepared for printing. A special case of the theorem has 
been rediscovered twice at least since 1873 ; the latest appearance 
being in Grunert’s Archiv^ Ixix. pp, 205-13, where the writer, 
K. E. Hoffmann, says regarding it:— ‘‘Diese allgemeine Formel 
enthalt nun alle in der Stern’schen Tabelle gegebenen als specielle 
Falle in sich, welche durch passende Wahl der etc, aus der 
