OF THE COMPOSITIONS OP NUMBERS. 
837 
This notion supplies, in consequence, a graphical proof of the theorem of Art. 2. 
Compositions of a number are conjugate when their graphs are conjugate. 
E.g. The conjugate graphs 
I 0-@------ -©----©-©-©- 
yield the conjugate compositions 
(214) (13111) 
The composition conjugate to a given composition may be written down, without 
constructing the graph, by the rules about to be explained. 
The composition must first be prepared— 
(i.) By writing successions of units in the power symbolism; for example, s succes¬ 
sive units must be written 1^; 
(ii.) By intercalating 1° between each successive pair of non-unitary parts; thus, 
when aa or ah occur {a and b being superior to unity) we have to write al’^a, al^h 
respectively. 
For the moment, call the non-unitary parts and the symbolic powers of unity the 
“ elements ” of the composition. 
When an element does not occur at either end of the composition it is called " non¬ 
terminal,” when at one end only “ terminal,” and when at both ends (thus consti¬ 
tuting the entire composition), “ doubly terminal.” 
The rules for procession to the conjugate are :— 
I. If m or 1”* be doubly terminal, substitute for m or m for U“. 
II. If m or 1”* be terminal, substitute l“~i for m or m + 1 for 
III. If m or 1“ be non-terminal, substitute 1““^ for m or m -j- 2 for 
The composition thus obtained is in the “ prepared” form and can be transformed to 
the ordinary form. 
E.g. To find the conjugate of (231141), take the “prepared” form 
(21031^41), 
and, beginning from the left, by 
Buie II. For 2 
substitute 
1, 
„ III. „ 1° 
2, 
„ III. „ 3 
J) 
1, 
„ III. „ r~ 
?? 
4, 
1—1 
1 — 1 
1 
5 ? 
„ n. 1 
35 
2, 
resulting in the conjugate composition, 
(12141"2), or non-symbolically (1214112) 
