of the Fifth Degree. 203 
tion of 6; for 
(57S)? —5?P (529) = 8. 
The object of research will not be a finite algebraic solution ; but 
I have ascertained (and it may be worth noticing) that the par- 
ticular form 
2° —5Q2?+2Q7Q?=0 
is soluble by radicals. 
97. The present discussion, then, seems to me to establish 
the insufficiency of two proposed methods of solving equations of 
the fifth degree, or rather equations in general, and to add to 
the moral evidence of the impossibility of the solution. Perhaps 
the want of universal assent to the argument of Abel may in 
some degree be owing to the want of uniformity in the views 
taken by Abel himself, by Sir W. R. Hamilton, by M. Kronecker, 
and by Galois, and it is unquestionably desirable that that argu- 
ment should be simplified. But I do not think that M. Wantzel’s 
modification of it meets the desire. The formule in his second 
step should, I think, be replaced by 
P (Hay Lay Wy yo + HH HM, Lay yy Lys s+), 
P(g, 21, Voy Ly -++)=ANG(La, My, M, Ly.--); 
P(@ ys Lay Bgy Ly+++)=AMP(tg, 2, yy Uy+- +), 
the only inference from which is 
aith+#=], or 1+X+m=0 (mod. 7), 
That n=8, AX=1, w=1 cannot, I think, be inferred without 
previously showing that the only prime power of an unsymme- 
tric function which can have two values only is a cube, and we 
are once more remitted to the arguments of Abel and Sir W. R. 
Hamilton. W. Wantzel’s third step, too*, seems open to objec- 
tion. Perhaps the impossibility of cubic radicals entering into 
the root may afford the basis of the desired simplification. 
4 Pump Court, Temple, 
February 6, 1860. 
* The cyclical interchanges of five do not coincide with the cyclical in- 
terchanges of three; and we can only infer that 
(ay = 62)’, 
where the symbols on the right refer to the quinary, and those on the left 
to the ternary interchanges. 
