THEORY OF THE PARTITION OF NUMBERS. 
640 
Permutations. 
Compositions. 
Partitions. 
y/iy/3ya 
(001 011 Oil) 
(00 01 02) 
yfiyafiy 
(001 011 111) 
(00 Of 12) 
/3 2 y 2 ay 
(022 101) 
(02 2 12) 
f3~ya.y 2 
(021 102) 
(02 12 2 ) 
/3y 2 a/3y 
(012 111) 
(OT 2 12) 
,6y 2 ay/3 
(012 101) 
(oT 2 IT) 
(3y 2 fiyu 
(012 Oil) 
(OT 2 02) 
fiyafiy 2 
(Oil Tl2) 
(of f2 2 ) 
(3y*y 2 ft 
(Oil 102) 
(of IT 2 ) 
/3yl3y z a 
(Oil 0l2) 
(01 02 2 ) 
y/3 2 yay 
(001 021 101) 
(do 02 12 ) 
y/3yay/3 
(001 011 101) 
(oo of IT) 
21 partitions; while for the enumeration, giving s 23 the values 0, 1, 2 in succession 
with Jc = 2, p = 1, q = 2, r = 3. 
= 3(1 XI X 0 + 2 X 2 X 1 + 1 X 3 X 1) = 3 X 7 = 21. 
The foregoing particular theory of the correspondence that exists between 
tripartite compositions and bipartite partitions is, for present purposes, sufficiently 
indicative of the general correspondence between (m -f- l)-partite compositions and 
a certain regularised class of m-partite partitions. 
§ 4. Constructive Theory. 
Art. 41. Given a line of route in a bipartite reticulation it may be necessary to 
enumerate the lines of route which lie altogether on either side of it. 
Thus in respect of the line of route delineated in the reticulation AB, lines of 
route exist which, throughout their entire course, are either coincident with it, 
MDCCOXCVI.-A.. 4 0 
