888 
MA.JOR P. A. MACMAHOX ON THE THEORY 
B 
C 
B 
A 
The process consists in writing down a unit for each terminal knot. The p — \ 
intervals between the units correspond to the p — 1 inter-terminal-knot spaces. If a 
space leads to a bifurcation in row B the symbol is + ; if in row (7 it is a blank 
sj)ace ; if in row Z) it is |. 
Thus, take the fifth tree above, we have 
1 + 1|1 
for the space a leads to a bifurcation in row B, giving the symbol -f- and the space h 
leads to a bifurcation in row D giving the symbol |. Hence the corresponding 
combination 
2| 1. 
On the same principle a tree can always be drawn to represent any combination of 
order k in respect of p units. 
70 . Passing to the case of multipartite numbers, consider tripartite numbers as 
representative of the general case. Arrange a row of multipartite units of the three 
kinds, viz. :— 
100 100 100 .. . 010 010 010 .. . 001 001 001 .. . 
and employing two different symbols, viz., the sign of addition and the blank space, 
we arrive at a certain composition of the tripartite P1P2P3, supposing the numbers of 
the units 100, 010, 001 that appear in the row to be p^ pj, and pg respectively. With¬ 
out altering the positions of the symbols introduced we may change the order of the 
units 100, 010, 001, and thus obtain other compositions. Permutations of these units 
between contiguous blank spaces are not permissible, so that for fixed positions of the 
pluses and blank spaces we do not obtain in general 
jPi + Ih + Ps) •' 
compositions, but some lesser number. 
This arises from the commutative nature of the symbol +. 
