131 
1X. — Researches in the Higher Algebra. 
By James Cocxte, M.A., F.R.A.S., F.C.P.S., 
Barrister-at-Law. 
Communicated by the Rry. R. Hartzy, F.R.A.S. 
Read October 5th, 1858. 
Sf. 
Tue higher algebra conducts us to results, the complex- 
ity of which is unfavourable, not only to the progress of 
research, but sometimes perhaps to intelligibility. The 
object of this paper is to exhibit some of them in a form 
which, without detracting from their real generality, will 
render their import comparatively easy of recognition. 
§ 2. 
Let i be an unreal fifth root of unity, and let 
i+=4(i), ?4+8=4(°), 84+ 4@)=4, 14246) =4, 
142¢(i)=c, 3+ 4()=d, 
then 
$() +o) =$@) -G@)= -1, 6+e=0, and 
ad=BP?=c?= — be=5. 
§ 3. 
Again, let v, w, 2, y and 2 be the roots of a given equa- 
tion in uw of the fifth degree, and let 
v+iw+?e+iy+iz=f(i), fd -f@ SE) -f ) =9 and 
(we + vy) +w (ve + yz) + 2(vz 4+ wy) + yew + #2) 
+2°(vy +we)=U=W(u), 
VOL. XV. te 
