253 
It is easy to demonstrate and construct, by a method 
similar to the foregoing, all M. Mathieu’s non-modular 
groups of (N‘+1)N‘(N‘ — 1), N being any prime number, 
without taking for granted any other group, and without 
having recourse to congruences ; but I do not think it worth 
the while to pursue the inquiry, as this method of combina- 
tions is certainly less general, although the separate cases may 
thus be made to appear more simple, than the methods which 
have been already given. The theory of combinations appears 
likely to owe more than it can contribute to that of groups. 
Two theorems are worth recording in combinations, which 
I owe to the study of groups. 
Eleven quintuplets can be formed to exhaust tioice the duads 
of eleven elements. This can be done by continual additions 
of 1 1 1 1 1 either to 1 2 3 5 8, or to 1 2 3 7 a (a=10). 
When N is any prime number , N 2 elements can be thrown 
into JN(N+1) (N— 1 )-plets, N+l N -plots, and |N(N— 1) 
(N+ \)-plets } so as once and once only to exhaust the duads 
of N 2 elements. 
This is proved by the systems of didymous radicals in the 
groups of (N+1)N (N — 1), when N is prime. All the triplets 
of these radicals are easily shown to be reducible to duads. 
In the Memoir of which the above is an abstract, this method, 
of combinations independent of equations, will be applied to 
other groups, superior and inferior to those here treated. 
The following Report of the Council was then read by 
one of the Secretaries : — 
In presenting the Annual Report, the Council congratulate 
the members on the improved condition of the Society, 
especially as regards its financial position. The balance in 
the Treasurer’s hands, as seen in his Report annexed, 
was on March 31st, 1861, £58. 2s. ; whilst on March 31st 
last, it amounted to £248. 6s. 7d . ; and this in spite of 
B 
