221 
then the quadratic equation becomes 
AEE 
and two unequal roots of it are the following: 
“a= el+t+ 7 —-&), 
g@2 = #(-1+7-j-A). 
Substitution and reduction give hence these two expressions : 
. 2n7r 
(2 e sin 
iy Fp balay yey cus 
3 
; an-1 ( os 
a+ ding 4 3 
ok fi 2m — la |. Que 
sin —— —— + J sm -3 
which may easily be verified by assigning particular values to 
n. No importance is attached by the writer to these particular 
results: they are merely offered as examples. 
5. It may have appeared strange that Sir William R. Ha- 
milton should have spoken of ¢wo unequal quaternions, as being 
among the roots, or two of the roots, of a quadratic equation in 
quaternions. Yet it was one of the earliest results of that cal- 
culus, respecting which he made (in November, 1843) his ear- 
liest communication to the Academy, that such a quadratic 
equation (if of the above-written form) has generally six roots : 
whereof, however, two only are real quaternions, while the 
other four may, by a very natural and analogical extension of 
received language, be called imaginary quaternions. But the 
theory of such imaginary, or partially imaginary quaternions, 
in short, the theory of what Sir William R. Hamilton has 
ventured to name ‘‘ Biguaternions,” in a paper already pub- 
lished, appears to him to deserve to be the subject of a sepa- 
rate communication to the Academy. 
