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 he 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 S 3 5 8, or to 1 2 3 T « («=10). 



When N is any prime number^ W- elements can he thrown 

 into JN(N-i-l) {^—X)'plets, N+1 1^-plets, and iN(N--l) 

 (^■\-\)-plets, so as once and 07ice only to exhaust the duads 

 of N^ 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. Ss. ; whilst on March 31st 

 last, it amounted to £M8. 6s. 7d. ; and this in spite of 



B 



