864 MAJOR P. A. MACMAHON OR THE THEORY 
To obtain compositions along these lines of route which have not zero-zero contacts, 
we have at disposal 
p + q — 26 -, 
p -\- q — 2s — I, 
p q — 2s — 2 
points respectively at which non-essential nodes may be placed, 
of compositions which have s positive-positive contacts (see Art. 
contacts is— 
Hence the number 
19) and no zero-zero 
IN 
which is 
(p + s) {q + s) fp\ M 
2Jq \sj \sj 
Hence the total number of compositions which have no zero-zero contacts is 
2p+,-2 (P + 1) (g + 1) ^ (p + 2) (g + 2) j -f- . . 
and, since the total number of compositions which have s positive-positive contacts is 
2P + 2- 
-1-. P\ ^ 
S J \s 
we find that the number of comjaosltions having s positive-positive contacts and also 
zero-zero contacts is 
2^ + 2 
-2.-2 J 2*+i _ (P + ^)(i + g) l fp\ fi 
pq 
s / \s 
and the total number of compositions having zero-zero contacts is 
2P+2-2 _j_ + I 2” _ 
(p + 1) (g + 1) 
pq 
r) (i) 
-f- . . . 
{p +2){q+ 2 ) | (q 
E.g,, putting qjq = 22, we thus verify 
4+ (4-1)4+ 1(8 -4) = 12 
that the number is 12. 
