198 The Rev. T. P. Kirkman on the Puzzle of 



aware, contains results which have gained Mr. Jerrard a great 

 and widely spread renown. But his undemonstrated proposi- 

 tion ought not to prevail against the well-considered argument 

 of M. Hermite. 



5. Mr. Cayley has calculated P. His value may, by means 

 of Mr. Barley's cyclical symbol, be written X'Xa^p (# a — #0)- 

 Developing this expression, and making the substitution 

 (•jP)(7 c )j we see that P 2 is a six-valued function. Hence, 

 by Lagrange's theory of similar functions, we find 

 E = r{P*}, S/(?«)=r{P>. ;) }. 



Speaking of my own personal convictions, I am satisfied that a 

 relation of the latter form ought to supersede the erroneous 

 result given in Mr. Jerrard' s ( Essay/ But the burden of 

 proof is on Mr. Jerrard, and he offers none. He simply (I say 

 it with a deference which long familiarity with his writings may 

 well inspire) intrenches himself behind a conjectural and inad- 

 missible equation, leaving others to explain how an equation 

 connecting the six and the twelve-valued function can exist at 

 all. The actual equation, I may add, does not give consistency to 

 the self-contradictory result which Mr. Jerrard has attempted 

 to verify, nor does it lead to an algebraic solution of the quintic. 

 Such a solution M. Hermite' s argument proves, on Mr. Jerrard' s 

 own premises, to be impossible. 



4 Pump Court, Temple, London, 

 February 3, 1862. 



XXX. On the Puzzle of the Fifteen Young Ladies. By the Rev. 

 T. P. Kirkman, A.M., F.R.S., Hon. Mem. of the Literary 

 and Philosophical Societies of Manchester and Liverpool*. 



MY distinguished friend Professor Sylvester, at page 371 of 

 the 21st volume of this Journal, volunteers en passant 

 an hypothesis as to the possible origin of this noted puzzle 

 under its existing form. No man can doubt, after reading his 

 words, that he was in possession of the property in question of the 

 number 15 when he was an Undergraduate at Cambridge. But 

 the difficulty of tracing the origin of the puzzle, from my own 

 brains to the fountain named (p. 371) at that University, is con- 

 siderably enhanced by the fact that, when I proposed the ques- 

 tion in 1849, 1 had never had the pleasure of seeing either Cam- 

 bridge or Professor Sylvester. My own account of the origin of 

 the problem may be seen at p. 260, vol. v., of the Cambridge 

 and Dublin Mathematical Journal, 1850. No other account of 



Communicated by the Author, 



