378 MAJOR P. A. AlACMAHON ON THE THEORY OF PARTITIONS OF NUAIBERS. 
section of the lattice is in the positive direction, markiug the successive points 
reached tivo, three, &c. 
For every point, marked s, vre have a factor, 
(a + c':) 
(-0 ’ 
and express the generating function as a product of I m such factors. If ?ibe co , each 
factor is of the form 
1 
(^)’ 
and if the number s appears a times on the lattice, ^ve have a factor and the 
complete result may be written 
_ 1 _ 
(.sp-i (.so)-Hs,r3... • 
Art. 95. Hence the enumeration is identical with that of the partitions of a 
unipartite number into an unlimited number of parts of + cr., + cr- + • • • different 
kinds, viz. 
a I of the numerical value but differently coloured. 
0”2 ’''2 5) 
O’s ,! )5 '^’2 >> 
'fhe number of distinct lines of route In a lattice of I in points is 
fl + m — 2\ 
\ 1-1 )’ 
so that, in general, on the lattice we have partitions of a number into I rn parts 
subject to ^ ^ descending orders. 
Such a partition is transformable {I S: m) into one composed of tlie parts 
1 of 1 colour 
rn 
I 
I + I 
111 
„ in 
,, m — 1 
I -f- m — 1 
1 
