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A Paper was communicated on July 1st, entitled, “ The 
Complete Theory of Groups, being the Solution of the 
Mathematical Prize Question of the French Academy for 
1860. By the Eev. Trios. P. Kirkman, M.A., F.R.S., and 
Honorary Member of the Literary and Philosophical Societies 
of Manchester and Liverpool.” 
I have the honour to present to this Society a complete 
solution of the problem proposed early in 1858 by the French 
Academy, as the subject for their Grand Prix dcs Mathe- 
matiques , for June, 1860. 
After the first contest of that date, at which something 
more was accomplished than what is common at a first com- 
petition on a question of the same difficulty, by the three 
candidates, of whom it was mv fortune to he one, it was 
withdrawn from further competition, without any award of 
the prize, with an unusual suddenness, which created some 
remark in the world of science. The reason assigned for this 
abrupt dismissal of a question, on which the savans of the 
imperial Institute had solemnly invited their scientific con- 
temporaries of all nations to undertake a two-years’ toil, was 
this, that while the Academy briefly thanked and commended 
all the competitors for contributing what was new and really 
important, on a theory whose complete discussion it had not 
hoped to obtain, that learned body was unanimously agreed, 
that its intentions about the unknown subject were not 
fulfilled. This is placed on record, without the faintest 
definition of those intentions, or publication of the facts of 
the contest ; the investigators receiving no definite credit, 
even in brief summary, for their numerous original results. 
Vide the curt report of the Referees, in the copious and 
weekly Comptes Rendus de V Academic, Mars, 1861. 
It is due to the question and to myself to state, that a letter 
was written by my friend Mr. Ekman, on my part, to the 
Proceedings — Lit. & Phil. Society— No. 2.— Session, 1863-64. 
