109 
450, as volumes of the Transactions of the Academy, and 
offering the fifty copies which still remained on hands at the 
same price as the former ones, namely, thirty shillings a copy. 
Reso.vep, on the recommendation of Council, that this 
offer be accepted. 
June 23, 1845. 
GEORGE PETRIE, ESQ,, Vice-President, in the Chair. 
James Strathearn, Esq., Daniel Conolly, LL. D., David 
Moore, Esq., Rev. Classon Porter, and James Talbot, Esq., 
were elected Members of the Academy. 
The following notice, by the President, Sir William R. 
Hamilton, of a theorem derived from his Researches on Qua- 
ternions, was read. 
oc 
4 
ad 
. 
Let AC’A’B’ be called a spherical 
parallelogram, if A’, B’, C’ bisect 
the sides BC, CA, AB of a spherical 
triangle ABC; and let it be said that 
the corner A of the triangle is the 
- point which completes the parallelo- 
gram when A’B’ and A’C’ are given 
as two adjacent sides thereof. 
Take any spherical quadrilateral, KLMN, and any point 
on the same spheric surface, P; draw the four ares PK, PL, 
_ PM, PN, and complete, in four points, K’, L’, M’, N’, the four 
_ spherical parallelograms, of which the given pairs of adjacent 
sides are PK, PL; PL, PM; PM, PN; PN, PK. Then 
p. the four new points, K’, L’, M’, N’, form a new spheric qua- 
 drilateral, such that its four sides, K/L’, L'M’, M’N’, N’L’, 
touch a certain spherical conic, having the poles of the dia- 
_ gonals KM, LN of the old quadrilateral for its foci. 
___ This theorem was stated to follow as an easy corollary from 
_ what Sir William Hamilton had already communicated to the 
_ Academy respecting quaternions. 
A 
