VIIL Memoir on the Tiieory of the Partitions of Numbers .—Part II. 
By Major P, A. MacMahon, R.A., D.Sc., F.Ji.S. 
Received November 21,—Read November 24, 1898. 
Introduction. 
Art. 64. 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 partition that is here brought into prominence. 
Let an integer n be broken up Into any number of integers 
if we ascribe the conditions 
the succession 
a„ ao, a.^, . . . 
tti > a. > ^3 > . . . > 
is what is known as a partition of n. 
There are s — 1 conditions 
> a.,, a., >: ois, . . . a,_i > 
to which we may add 
a, > 0 
if the integers be all of chem [)ositive (or zero). For tlie present all the integers are 
restricted to be 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 
> > >» 
< a._> < a3 . . , < 
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 
3 —3 
5.5.99 
