116 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. 



when deprived of the terms linear in p lt p 3 , ..., and of the term independent 



For it is easy to show that two consecutive terms 



N , in t I in\ T> ./ \t+i m t 2 / n 



, y m-t-n-p , 



~^rW ""'- 1 



may be given the form 



, \ t m t 1 fn\ /pi+mtl\"' ip 2 



~ 



+mtI\'' 



,( 



Pl p, 



, w + m-<-2/n \ fpi+m t-3\'< (p 3 +m 1-3\'* 

 H m _l -(t + l)( Pl p 2 



and, giving t the values 0, 2, 4, ..., and summing and simplifying, we obtain 



ipi 



! + m 1 \" ipi + m - 1 

 \ 



Pi \ P* 



(n+l\ 



2 



v 



which we know to be the value of N m . 



Art. 76. The symmetry of the numbers N m _/ will not escape the notice of the 

 reader. 



SECTION 9. 



Art. 77. My purpose now is to connect the preceding pages with my Memoir on 

 the Compositions of Numbers, to which attention has already been directed. In the 

 course of that investigation I had occasion to consider the permutations of the 

 letters in a?p>y, 



with the object of determining the number of permutations containing given 

 numbers of a contacts, 



If we take any permutation 



,../Ja...ya...y/8...y/3a... 



and particularly notice all of such contacts, it is clear that the numbers of parts 

 in the descending specification a, j3, y, ..., being numbers in descending order of 



