S96 Proceedings of the Royal Society of Edimhurgh. 
. V V V V 
Then a = sin. a (1 + - -f- — ^ + — tt, -i -f- &c.) 
n nn' 7in' n" n n n n" ' 
The expressions for elliptic or hyperbolic areas, and for the 
logarithm of any number, are exactly of the same form. 
The terms of this expression are formed from the cosines of 
a series of arcs or sectors, which constitute a geometrical pro- 
gression. In like manner, the terms of the expression for an 
elliptic or hyperbolic sector, are formed from a series of abscis- 
sas corresponding to elliptic and hyperbolic sectors, each of* 
which is one-half of that before it. These abscissae are found 
by precisely the same formula in the two curves ; and in the 
beginning of the paper, general theorems are investigated, 
which express the relation between the co-ordinates correspond- 
ing to an elliptic or hyperbolic sector, and those which cor- 
respond to any multiple of that sector. The theorems are de- 
duced from a single property common to both curves, without 
employing any geometrical constructions, and without introdu- 
cing impossible quantities. 
Mr Thomas Allan read an account of a Petrifaction found 
in the neighbourhood of Edinburgh. 
At the same meeting, a paper by Mr Hercy was read. On 
the effects of injecting a solution of Opium into a Vein, in an 
anomalous nervous affection. 
At the same meeting the following members were elected ; 
Honorary Member. 
Count Berthollet. 
Foreign Members. 
M. Vauquelin. 
M. Legendre. 
M. Brochant. 
Baron Von Buch. 
M. Berzelius, 
Baron Krusenstern. 
M. Sismondi. 
M. Poisson. 
M. Prony. 
M, Gauss. 
M. Blumenbach, 
Count Volta. 
M. Kaussler. 
M. Degerando. 
Ordinary Members. 
John Hay, Esq. younger of Hayston. Dr Ballingall. 
Captain Robert Hay, R. N. 
The Society adjourned till November next. 
