858 
MAJOR P. A. MACMAHON OR’ THE THEORY 
letters a may occur with each distribution of the letters Hence the total number 
of permutations is 
24. Third Proof.^ 
I now come to a method which is valuable in the theory of multipartite numbers 
in general, and also intrinsically of great interest. 
I consider, as in the second proof, the permutations of the 2^ + 2' letters 
wdiich exhibit exactly s, /3a-contacts. The proof depends upon showing there exists 
a one-to-one correspondence between these permutations, and those in which the 
letter ^ occurs exactly s times in the first id places counted from the left. 
Suppose the permutation with s, ^a-contacts to be 
23a y8a /8a a^*/8''‘ /3a a^®y8^® /3a a^®/8’^® 
where, for convenience, s has the special value 5. 
Any of the indices x, y may be zero. 
Observe that 
+ ^3 + + ^5 -h ^6 + 5 = p, 
//i + //3 + 2/3 + //r + 2/5 + 2/6 + 5 = ^. 
Obliterate the letters which do not occur in /3a-contacts, and the letters a, 
which do occur in /3a-contacts, there remains a succession 
a*i/3a^®/3a^®^a^^/3a'^®y8a^® 
of ]D letters, /8 occurring s times. 
N ext obliterate in the original permutation the letters a, wdiich do not occur in /3a- 
contacts, and the letters /3, which do so occur; there remains a succession 
*/8‘"'a^^^a/3^®a;8'"*ay8'"®ay8'"® 
of q letters. 
Take these two successions for the left and right portions of a new permutation, 
viz.— 
a^'*/8a^®/3a^®/3a''^y8a^®/8a®® . /3"’a/8'"®ay8^®a/3-'*a/3^®ay8^®a, 
and we have made a perfectly definite transformation of a permutation involving 
exactly s /8a-contacts into another possessing the property that the letter /8 occurs 
s times in the first p places. 
E.g., to transform 
/8^a®y8^a, 
write it 
(PjP /3a a/8^ /3a; 
* This proof is of fundamental impoi’tance in the succeeding part of this investigation. 
