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, 
pt y-l p> 
where p and p’ denote two geometrically real vectors, and / —1 
is the old and ordinary (or commutative) imaginary of common 
algebra, and generally of biguaternions such as 
g+v-l1dq; 
where g and ¢ 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theory will be found in his (as yet unpublished) Lectures on 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 with null squares, 
such as 
t+ hj, or f+ hk, 
where i, j, 4 are the peculiar symbols of the quaternion caleu- 
lus, observing the laws communicated by him to the Academy 
