4 
“ On the Rev. T. P. Kirkman’s Method of Resolving Alge- 
braic Equations.” By the Rev. Robert Harley, F.R.S., 
Corresponding Member of the Society. 
In two papers published last March in the Proceedings of 
this Society (vol. vii, pp. 183-7, 141-8), the Rev. T. P. Kirkman 
proposed a method of resolving algebraic equations of all 
degrees. A few weeks ago I pointed out to Mr. Kirkman in 
a private communication that his method, when applied to 
the higher equations, fails. I objected in particular that in 
dealing with the general quintic he assumes that certain 
functions which are only symmetrical in four of the roots, 
are symmetrical in all five, and that this false assumption 
vitiates his solution. Mr. Kirkman has since acknowledged 
(vol. vii, p. 222) in the most full and sufficient manner the 
validity of my objections. 
I find by a letter received yesterday that my distin- 
guished friend the Chief Justice of Queensland has taken 
substantially the same view of Mr. Kirkman’s method as 
myself. His remarks are so interesting and instructive 
that I venture to lay them before the Society, I take the 
liberty also to append thereto some remarks of my own : 
The Chief J ustice of Queensland to the Rev. R. Harley. 
“ Brisbane, Queensland, Australia, 
“July 30, 1868. 
“I am not disposed to assent to Mr. Kirkman’s views 
on the quintic, i. e., not to the validity of his alleged 
solution. His assumed form, at p. 134 of the Manch. Proc., 
vol. vii., is not the same as Lagrange’s. Compare Serret 
Gouts, 2nd ed., p. 567. Let us put 
— Xq + ClX^ + Cl -r Cl X 3 -f- Clb/4, 
then the Lagrangian theory gives 
x h = | 'ZiX 4- a~ k (a,x) + a~ 2k (ci 2 ,x) + a~ 2k {cd,x) + a~ 4A ’(ci 4 ,#) | 
which seems to be different from Mr. Kirkman’s formula. 
As I understand it, the theory of the resolvent is this, — 
