6 
for you to see, and also for you to compare Mr. Kirkman’s 
system of permutations, at p. 141 (ib.), with that in my 
“ Notes on the Higher Algebra,” Q. J., vol. iv., p. 53, and to 
turn to my “ Supplementary Researches,” &c., in the 
Manch. Memoirs. In forming the system 
Hf+ &c. =(A % . . . H 2 \+ &c. =(F )i (ib., p. 142), 
Mr. Kirkman has only taken account of permutations of 
%i,x 2 ,x 3 , and x±. He takes no account whatever of the 
permutations of x 0 , further than observing (ib., p. 142) H x 
is invariable for the cyclical permutation of x 0 ,x x ,x§x 3 ,xg He 
neither asserts nor denies that H 2 , H 3 and H 4 are invariable 
under that permutation. I suspect that it will ultimately 
appear that Mr. Kirkman has only succeeded in expressing 
H x , &c., as functions of x 0 , and not in terms of symmetric 
functions of all the roots. And functions of x 0 are useless 
for the purpose of solving the quintic. You speak of Mr. 
Kirkman’s presenting a solution of the general equation of 
the 7th degree. I think you are right in believing that he 
is quite wrong. Probably the same error runs through it as 
in his paper on quintics and sextics. Even if Mr. Kirkman 
has not solved the quintic, still if he has made the solution 
of a sextic depend upon that of a quintic, that will be an 
advance in the theory of equations. Rut I am as little able 
to assent to Mr. Kirkman’s theory of sextics as to his theory 
of quintics — in which latter, I may add, there is an internal 
variance ; for at p. 134 he seems to derive H 1 ,H 2 ,H 3 ,H 4 
from the group 
01234 (H,) 
02341 (H 2 ) 
03412 (H 3 ) 
04123 (H 4 ) 
while at p. 141 he seems (compare p. 142) to derive those 
four H's from the group 
1234 
2143 
3412 
4321 
