388 



Sir William R. Hamilton, LL. D., gave an account of the 

 geometrical interpretation of some results obtained by calcula- 

 tion with biquaternions. 



In this communication bivectors were employed, and were 

 shown to conduct to interesting conclusions. The conception 

 of such bivectors, 



p + y - 1 P \ 



where p and p denote two geometrically real vectors, and \f - 1 

 is the old and ordinary (or commutative) imaginary of common 

 algebra, and generally of biquaternions such as 



q + V - 1 q', 



where q and q are real quaternions, interpretable geometrically 

 on the author's principles, had occurred to him many years 

 ago ; and the remark which he made to the Academy in No- 

 vember, 1844 (see the Proceedings of that date), respecting 

 the representations, in his Calculus, of the geometrically un- 

 real tangents to a sphere from an internal point, as having 

 positive squares, belonged essentially to this theory of bivec- 

 tors. In the same year, the more general theory of biquater- 

 nions had occurred to him, in connexion with what in his 

 theory presented themselves as the imaginary roots, or purely 

 symbolical solutions, of a certain quadratic equation in qua- 

 ternions. Notices on the subject have since appeared in his 

 subsequent papers, in the Proceedings of the Academy, and 

 in the Philosophical Magazine : and a fuller statement of the 

 theory will be found in his (as yet unpublished) Lectures on 

 Quaternions, of which many sheets have long since been dis- 

 tributed among his friends and others in the University. On the 

 present occasion he has employed bivectors ivith null squares, 

 such as 



i + hj, or j + hk, 



where i, j, k are the peculiar symbols of the quaternion calcu- 

 lus, observing the laws communicated by him to the Academy 



