L <* J 



II. Second Memoir on thr Compositions of \ttmbtrs. 

 By Major P. A. MAcMAHOx, R. A., D.Sc., F.R.& 



Received August 23, Read December 6, 1906. 



PREAMBLE. 



bi a Memoir on the Theory of the Compositions of Numbers, read before the Royal 

 Society, November 24, 1892, and published in the 'Philosophical Transactions' for 

 1893, I discussed the compositions of multipartite numbers by a graphical method. 

 The generating function produced by the method was of the form 



a symmetrical function of the quantities a. 



The investigation of the present paper leads, in part, to the same generating function 

 which is subjected to a close examination. Moreover, the whole research has to do with 

 the compositions of numbers, and appropriately follows the Memoir of 1 893. 



The problem under investigation, which was brought to my notice by Professor 

 SIMON NEWOOMB, may be stated as follows : 



A pack of cards of any specification is taken say that there are jt cards marked 1. 

 <l cards 2, r cards 3, and so on and, being shuffled, is dealt out on a table ; so long at 

 the cards that appear have numbers that are in descending order of magnitude, they 

 are placed in one pack together equality of number counting as descending order- 

 but directly the descending order is broken a fresh pack is commenced, and so on until 

 all the cards have been dealt. The result of the deal will be m packs containing, in 

 order, a, b, c, ... cards respectively, where, n being the number of cards in the whole 



(ate...) 



composition of the number ?, the numbers of parts in the composition being m. 

 We have, then, for discussion 



(1) The number of ways of arranging the cards so as to yield a given composition 



(ate...); 



(2) The number of arrangements which lead to a distribution into exactly m packs. 

 These problems, and many others of a like nature, are solved in this paper. 



VOL. OCV1L A 414. K 21.1.07 



