72 
rence in the quotients obtained for one root is inverse of that for the 
other root of the quadratic ; second, that in continued fractions 
which express the square root of any rational fraction, the order of 
recurrence is symmetric within each period ; and third, that the 
last quotient of the period must be double of the first or integer part 
of the root. 
It was also observed, that the progressions may be separated into 
as many progressions as there are members in the period, each 
progression proceeding as with a single repetend. The considera- 
tion of this branch of the subject was said to lead to the formation 
of series representing the impossibility of the roots when the square 
of the middle coefficient is less than four times the product of the 
extreme ones, and when these have the same sign. 
The following Gentleman was duly elected a Fellow of the 
Society : — - 
Frederick Field, Esq., Chili. 
The following Donations to the Library were announced : — 
Scheikundige Verhandelingen en Onderzoekingen uitgegeven door 
G. J. Mulder. Eerste deel, derde stuk. Het Bier scheikun- 
dig beschouwd door G. J. Mulder. Rotterdam, 1857. 8vo. 
— From the Author. 
Flora Batava, 182 Aflevering. Amsterdam. 4to.-~ -From the King 
of Holland. 
A Catalogue of 3735 Circumpolar Stars, observed at Bedhill in the 
years 1854, 1855, and 1856, and reduced to mean positions 
for 1855*0. By Bichard Christopher Carrington. London, 
1857. Folio. — From the Lords of Admiralty. 
Denkschriften der Kaiserlichen Akademie der Wissenschaften. 
Mathematisch-Naturwissenschaftliche Classe, Zehnter und eilf- 
ter bande. — From the Vienna Academy. 
Annales de l’Observatoire Physique Central de Russie. Par A. T. 
Kupffer. Annee 1854. St Petersbourg, 1856. 4to. — From 
the Observatory. 
Bulletin de la Societe Yaudoise des Sciences Naturelles. Tome v.. 
No. 41. Lausanne, 1857. 8vo. — From the Society. 
Proceedings of the Natural History Society of Dublin, for the Ses- 
sion 1856-57- Dublin, 1857. 8vo. — From the Society. 
