380 ]\[AJOE P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
/>., denoting the .s^’’ figurate number of the second order, and h, — — hs_y,^ is easily 
shown to be equal to the number of points of the lattice marked s. We liave to 
show that this is equal to 
.. = iU (/ + ,,,) 
II -—-• 
.= 1 0) 
Taking- / > m. ol^serve that (/ + •''■) occurs in tlie former to the power 
which 
= 1 if / -}- .s > m 
= 0 if ? -T .S' < rn ; 
whilst (s) occurs to the power 
hs’-i 
wliich 
— I it S > HI 
= 0 if .S' >. I and < /a 
= — 1 if 5' < / ; 
the product is, therefore, 
{(/ + T) (/ + 2) ■ ■ . (m)] 0 Ifm + 1) (m + 2) ... {I + i/i)} _ (I +_y) 
{(1) (2) . . . (/)} {{I + 1) (/ + 2) . . . (m)] 0 ' ” (.) ' * 
Art. 97. When I =: m =■ n = co the generating function is 
(1 _ .,) (1 _ (1 _ (1 _ . . . 
which may be written 
o__ 
(1 — ff]A-) ( 1 
C., H. 
d' . 1. - - 
ffo 
1 _ -A ^,.1 _ 1 _ lA W. . .) 
a- 
a. 
from which is deduced a graphical representation in two dimensions involving units 
and zeros. 
The gi'apli is regular, and the successive rows involve the numbers 
1 ; 1 , 0 ; 1 , 0 , 0 ; 1 , 0 , 0 , 0 ; . . . 
re.spectivehn In the general case there is a similar representation, jiroper restrictions 
being placed upon the numbers of figures in the rows, 
