PROCEEDINGS 
OF 
THE ROYAL IRISH ACADEMY. 
1844-45. No. 48. 
November 11, 1844. 
SIR Wm. R. HAMILTON, LL.D., President, in the 
Chair. 
His Grace the Duke of Leinster, Most Noble the Marquis 
of Downshire, Baron Farnham, and Baron Wallscourt, were 
elected Members of the Academy. 
The Chair having been taken, pro tempore, by the Rev. 
J. H. Topp, D.D., V.P., the President gave an account of 
some additional researches in the theory of Quaternions, or of 
a new system of Imaginaries in Algebra. 
In the theory which Sir William Hamilton submitted to 
the Academy in November, 1843, the name guaternion was 
employed to denote a certain quadrinomial expression, of 
which one term was called (by analogy to the language of or- 
dinary algebra) the real part, while the three other terms 
made up together a trinomial, which (by the same analogy) 
was called the imaginary part of the quaternion: the square 
of the former part (or term) being always a positive, but the 
square of the latter part (or trinomial) being always a nega- 
tive quantity. More particularly, this imaginary trinomial 
was of the form ix + jy + kz, in which 2, y, z were three real 
and independent coefficients, or constituents, and were, in se- 
veral applications of the theory, constructed or represented by 
VOL, III. B 
