161 
Ordinary Meeting, October 20th, 1863. 
E. W. Binney, F.R.S., F.G.S., President, in the Chair. 
A Paper hy the Rev. T. P. Kir km an, M.A., F.R.S., “ On 
the Complete Theory of Groups,” was read. 
The following communication was also made by Mr. 
Kiukman : — 
I observe that I have not accurately expressed my meaning 
at page 138, art. VI., of the Proceedings of this Session 
The theorem, which can hardly he new, is correctly enun- 
ciated thus : — 
Theorem. If 0 o and <j> a be two substitutions of any order p, 
and of the signature a , in any group T, and if 
H = 1 + 0„ + 0 - + . . 
contains no group of powers, h, permutable with 
F = 1 + <j) a + (j>a 4- . . 
F being different from H, T cannot contain, as A a , fewer 
than R p (2-f- R p ) substitutions having the signature a ; R p 
being the number of integers, unity included, which are less 
than p and prime to it. 
The proof is simple. Let 6 h 9 . . he the R^ principal 
substitutions of H. Then the 2+ R,, groups 
F, H, O b FQ- b , 6> C F0- C . . . 
are all different, unless either 
F = 0 e F0~ e , 
or 
d c F0- c = 0^F0- f , 
6" and 6 f being principal. 
Proceedings — Lit. & Phil. Societt— No. 4.— Session, 1863-64. 
