IN THE HIGHER ALGEBRA. 137 
2°. Since Sir W. Rowan Hamilton’s “ Inquiry into the 
Validity of Mr. Jerrard’s Method” has established the 
fact that the general equation may be reduced to the form 
(1), that form imposes only a seeming, not a real, restric- 
tion on the generality of our investigations. 
3°. The same remark applies, when C is unrestricted 
in value, to (13); for to such a form the general equa- 
tion of the sixth degree may be reduced by Mr. Jerrard’s 
method. 
4°, The condition (14) appears to restrict the general- 
ity of the sextic. But the solution of the restricted sextic 
is reduced to that of (1), or, which amounts to the same 
thing, the solution of (15), (16) or (17) is reduced to 
that of (18.) 
5°. Were it possible to solve (15), (16) or (17) by 
means of equations of degrees lower than the fifth, we 
should probably be advancing a step towards clearing up 
the difficulties which encircle the question of the finite 
algebraic solution of (1), and, therefore, of the general 
quintic. 
6°. If the roots of (15), (16) or (17) are connected, 
symmetrically or otherwise, with those of the final sextic 
in a proposed method of solution of a quintic, the solvi- 
bility or insolvibility of those equations may afford a test, 
perhaps the simplest which the case admits, of the suf- 
ficiency of the proposed method. 
§ 19. 
Mr. Harley, in whose hands I was so fortunate as to 
place these investigations, has since verified all the co- 
efficients of the equation in 0, with the exception of the 
last two. He has effected this by a direct, original and 
general process, of independent interest and of intrinsic 
value and importance. His results, perhaps carried still 
further, will, I hope, be soon laid before the Society. 
