224 Major P. A. Macmahon. 



" Memoir on the Theory of the Partitions of Numbers. Part II." 

 By Major P. A. MACMAHON, R.A., D.Sc., F.R.S. Received 

 November 21, Read November 24, 1898. 



(Abstract.) 

 Introduction. 



The subject of the partition of numbers, for its proper development, 

 requires treatment in a new and more comprehensive manner. The 

 subject matter of the theory needs enlargement. This will be found 

 to be a necessary consequence of the new method of regarding a parti- 

 tion that is here brought into prominence. 



Let an integer n be broken up into any number of integers 



a l a 2> a 3> ...... a > 



and we ascribe the conditions 



a ! 5a 2^ a 3~" ...... a *> 



the succession 



is what is known as a partition of n. 

 There are s-l conditions 



to which we may add 



*s:0, 



if the integers be all of them positive (or zero). 



For the present all the integers are restricted to the positive or zero 

 by hypothesis, so that this last-written condition will not be further 

 attended to. 



If, on the other hand, the conditions be 



no order of magnitude is supposed to exist between the successive parts, 

 and we obtain what has been termed a " composition " of the integer n. 

 Various other systems of partitions into s parts may be brought 

 under view, because between two consecutive parts we may place either 

 of the seven symbols 



We thus obtain 7 s l different sets of conditions that may be 

 assigned ; these are not all essentially different, and in many cases they 

 overlap. 



