OF THE COMPOSITIONS OF NUMBERS. 
887 
§ 5. Extension of the idea of Composition. 
68. The idea of composition is capable of enlargement from a particular point of 
view. 
In regard to unipartite numbers, consider^ units placed in a row 
111111 ..., 
there are p — 1 spaces between them which may be occupied by algebraic symbols at 
pleasure. We may select a definite number of different symbols, and choose any one 
to occupy any one of the p — 1 spaces. If this definite number be h, we can, by 
filling each space at pleasure, arrive at different expressions involving the p 
units. 
Clearly we may take as one of these symbols the simple unoccupied blank space,* 
since such space left between any two numbers, quantities, or expressions of any 
kinds has, in every case, a well-understood signification, not always, however, the 
same, in mathematical work. 
If we restrict ourselves to a single symbol, only one expression involving the 
units is possible ; choosing this symbol to be that which indicates addition we 
merely get 
which is p, or the number which enumerates the units. Had the chosen symbol been 
that denoting subtraction, the expression would have denoted — (p — 2); a blank 
space would have yielded a succession of p units; the symbol of multiplication, unity, 
and so forth. 
All the modes of obtaining expressions from the p units may be called combinations 
of the first order in respect of the p) units. 
Passing to the case of two different symbols we may choose to employ the sign of 
addition and the blank space. We thus obtain 2^“^ different expressions which are 
the several compositions of the number p. 
In general, expressions obtained by choosing from any h, different symbols may be 
called combinations of order h in respect of the p fundamental units. 
69. There is a one-to-one correspondence between these combinations and the 
“ Trees ” which have an altitude h and p terminal knots. 
As an example take k andp) each equal to 3, and, further, take as symbols, the sign 
of addition, the blank space and the symbol | of unspecified signification. 
The correspondence between the nine trees and the nine combinations of order 
three is shown below : — 
* The unoccupied blank space might be represented by a definite symbol such as O, as suggested to 
me by one of the referees. 
