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. 

 Resolved, 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. 



Let AC'A'B' be called a spherical 

 parallelogrmn, 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 K'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 arcs 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 

 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. 



