141 
Ordinary Meeting, March 31st, 1868. 
Edward Schunck, Ph.D., F.R.S., &c., President, in the 
Chair. 
“ On the Solution of Algebraic Equations,” by the Rev. 
Thos. P. Kirkman, M.A., F.R.S. 
I showed in the preceding number of these Proceedings that 
if x 0 x x x 2 x 3 x x are the roots of the quintic F=0, we can write, 
where S is a symmetrical function of the roots, and Eh H 2 H 3 H 4 are 
four values of a 24-valued function of those roots. I can now 
show how to form these four values with the coefficients of F=0 ; 
i.e. how to solve the famous quintic by a method which leads 
directly to the solution of all algebraic equations. We write the 
complete group of 24 substitutions among four elements thus : 
1234 
1342 
1423 
1243 
1324 
1432 
2143 
3124 
4123 
2134 
3142 
4123 
3412 
4213 
2314 
4312 
2413 
3214 
4321 
2431 
3241 
3421 
4231 
2341 
A 
B 
C 
D 
E 
F 
The 12 substitutions over ABC are all what we call positive , 
that is, they are made by an even number of transpositions; the 
12 over D E F are all negative , that is, made by an odd number of 
transpositions. Thus 1243 is made by the transposition of 4 and 
3; 2341 by the three transpositions 21, 31, 41. The systems 
(A+B+C) and (D-J-Eq-F) are changed one into the other by any 
odd number, and remain both unchanged by any even number, 
of transpositions. 
The entire number of substitutions possible with r elements 
can be written down in like manner in a sum of two systems, 
Pboceedings— Lit. & Phil. Society— Vol. VII.— No. 12.— Session, 1867-8. 
