3G0 MAJOR P. A. MACMAHOR" ON THE THEORY OF PARTITIONS OF NUMBERS. 
wlierein putting 
Xi = X 3 = . . 
Yi Yo = . . 
. = X. = X, 
• = Y, = 2 /, 
we obtain the enumerating function 
. 1 _ + ... 1 . . . + T)^X'i+/i'i+. . . +I)'i ’ 
in which we seek the coefficient of x'"y'^'. 
Art. 71. Ex. gr. Consider the particular case 
Oti *^2 “h • • • “h ~~ 
otj -]— 2ot2 “h • • • “h ^ J 
«!, an,. . . a^ being, as usual, subject to the conditional relations. 
The enumerating function is 
_ 1 _ 
1 — xy . 1 — xhf. 1 — ... 1 — x’y^’' ’ 
and it is obvious also that the partitions of the bipartite ww' which satisfy the 
conditions may be composed by the biparts 
11, 23, 36, . . . i, ^iii+l). 
The corresponding graphical representation is not by superposition of lines of nodes, 
but bv angles of nodes, of the natures 
i/ c5 ' 
(J 9 
' 3 
0 
0 
0 
0 
0 
0 
o 
Art. 72. It is convenient, at this place, to give some elementary theorems concerning 
the 12 function which will be useful in what follows. 
12 —- =- 
> 1 1 — x.l — xy ’ 
\ — ax .1 — — y 
a 
n 
1 — ocyz 
\ — ax. 1 — ay .1 — — 
a 
1 — x.l — y .1 — xz.l — yz’’ 
