Algebraic Solution. ; 383 
the gratitude of mathematicians, it is scarcely possible to ignore 
the fact that Mr. Jerrard’s hope (expressed at the conclusion of 
his paper of 1845), to discuss the resolution of the trinomial 
equation z°+ A,v#+A,=0, has not been realized, and that little 
or no approach has yet been made towards its realization. Mr. 
Jerrard’s subsequent researches on quintics seem to me, for rea- 
sons already adduced, to enhance rather than dispel any diffi- 
culties which arise upon the paper in question. It is perhaps 
to be desired that the mathematical world should be made ac- 
quainted with the whole of Mr. Jerrard’s views on this import- 
ant subject. 
Does an absolutely impossible, or rootless, equation, exist ? 
MM. Terquem and Gilain have discussed this question in the 
Nouvelles Annales de Mathématiques*, with reference ‘to the 
equation 
+V1+2e4+ V1—2=1. 
But this equation does not in reality raise the question under 
consideration. Jor (as I had occasion to write to Mr. Harley 
durmg last autumn) every one of the congeneric equations 
is soluble. And some one of the four values of a given by 
e= +(+1)?3./3 
will satisfy any one of the above four congeners. I shall there- 
fore again (S. 3. vol. xxxvii. p. 281) have recourse, for illustra- 
tion, to the equations 
1+ Vz—44+ Vz—1=0, 
1— Vx—4— Wx—1=0, 
each of which must, I think, be deemed impossible or rootless. 
The only gleam of a solution of the last is, so far as I can see, 
one which springs from the assumptions 
v=4(+1)?4+(—1)?, 
A= 4( tg De; l= Ss. 
while, for the first, we have the system 
#=4(—1)? +(+))?, 
4=4(—1)?, T=(+1)* 
But how can that be called a solution which depends upon a mo- 
dification of the constants of a problem (compare 8. 4. vol. ii. 
p. 439)? The safer conclusion seems to be that the two equa- 
tions are rootless. 
Midland Circuit, at Lincoln, 
March 15, 1861. 
* See Mr. Wilkinson’s Note Mathematice, Mechanics’ Magazine, 
vol, lxii. p. 582. 
