860 
MAJOR P. A. MACMAHOX ON THE THEORY 
compositions, because the p q — s — \ non-essential nodes on each line of rout« 
may be selected as composition-nodes in this number of ways. 
Hence, the total number of compositions is 
F {pq) = t 
{cf. the expressions for this number obtained in Arts. 12 and 14). 
Analytically, this result is equivalent to the algebraic expansion 
1 _ 1 ■ 2xy , 
X-^ix^y-xy) (1 - 2 ^) (1 - %j) (1 - 2 ^-)^ (1 - 2yf (1 - (1 - 2xjf ‘ • 
26. The important transformation of permutations established above is interesting 
when viewed graphicaliy. 
A line of route is traced by a certain succession of a and /3 segments. That 
portion of a line of route traced by the initial p segments terminates in one of the 
points 0, 1, 2, 3, 4 in the diagram. All these points lie on a straight line through 
the right-hand lower corner. If one of these points be marked s, s of the p segments 
will be yS segments, and every line of route passing through the point s has the 
property that s symbols yS occur in the first p places of the corresponding permu¬ 
tation of the symbols a and y(3. Hence, there is a one-to-one correspondence between 
the lines of route possessing 0, 1, 2, 3, 4, . . . s, . . . essential nodes and the hnes of 
route passing through the points marked 0, 1, 2, 3, 4, . . . s, . . . 
Observe that the number of lines of route in the graph, of which A and s are the 
initial and final points, is (^), and in the graph of which s and B are the initial and 
final points, is 
Hence of the grapli AB the number of lines of route passing through the point s is 
and since every line of route must pass through one of the points 0, 1, 2, 3, 4, ... 5, 
. . . we have 
