MAJOR P. A. Mv.-MAIlnX OX THE COMPOSITIONS OF NUMBERS. 117 



magnitude, is necessarily one greater than the numtar of such contacts ; in the 

 present instance there are 6 parts in the descending specification and 5 contacts. The 

 problem of the determination of the pmiiutiitions having descending specifications 

 containing m parts is identical with that which is concerned with those lmvingm-l 

 contacts of the nature specified. 



Art. 78. I established in the Memoir that the letters in 



can be permuted in 



\ n J \* 

 ways so as to have exactly ^ ^ contacts> 



% 7* 



> y 

 and I further discovered that this number is the coefficient of 



in the development of the function 



(a + A^jS + Any)' (a + + A^y)' (a + ft 4 y )'. 



Art. 79. In the same paper I showed that for this function may be substituted 

 the function 



which does not involve p, q, r, and may therefore lie regarded as the general 

 generating function of the numbers. 



Art. 80. Reserving for the present the generalizations, which were also given in 

 the papers referred to, it is clear that the application to the present question is 

 obtained by putting 



AJI = Aai = Asa A > 



when we find that the number of permutations of 



>V, 

 which have descending specifications containing m parts, is the coefficient of 



in the development of ( 



or of 1 



I - (a + + y) + ( 1 - A) (a/3+~ay +y)-(l - A^y 



This, therefore, is the true generating function of the numbers N. 



