Fimctions given by Groups. 3 1 5 



On the number and formation of many-valued Functions 

 of X1X2X3 — x„, which of any degree can be con- 

 structed upon any Group of those elements, with 

 exhibition of all the values of the Functions. By 

 Thos. P. Kirkman, M.A., F.R.S. 



{Received May 8th, iSgi.) 



I. My object in what follows is a double one. One aim 

 is to communicate a new theorem of remarkable power in 

 the search of many-valued functions of n letters, which adds 

 an elegance, though nothing of rigour or completeness, to 

 my solution worked out over thirty years ago, of the Prize- 

 Question proposed at Paris early in 1858 for the competition 

 of i860. 



Another aim is to expand, so as to make them more 

 intelligible to readers not supposed to have seen their 

 way through the preceding chapters, and who know only 

 the simplest elements of the Theory of Groups, the two 

 pages, 342 and 343, of the treatise On Groups and many- 

 valued Functions y which this Society did me the great 

 honour to print at once, in Vol. I. Sen 3, 1862, of its Memoirs. 

 To that treatise I shall refer by M.M. ; and I hope to 

 convince the reader, who knows how to define a substitution 

 and a group, and who can perform the operations in 

 substitutions, 



AB = C, BA-D AA = A2, &c., 



{M.M. p. 275), that in the propositions of those two pages 

 is given a complete and demonstrated answer to the above- 

 mentioned Prize-Question of the Institute. 

 T 



