MAJOR P. A. Mw-MAHoX o\ THE COMPOSITIONS OF NUMBERS. 123 



The. ExpressilnUty of D,. 

 Art. 95. The fundamental relation 



exhibits D, in terras of powers of DI. 



It is clear, A pinori, that D, is expressible in terms of D, and powers of D,, e.g., 



) (3) D = D ' ( D ' Dl + 8W) + 96 ' Dl + 1 2fti ) 

 (2) (4) Da = D a( D ^+^I ) ,D I -f-29^D,+24i 3 D 1 + 306 4 ), and so on, 



where notice, as a verification, that the siim of the numerical coefficients is the same 

 on the two sides. 



In every case D s appears as a factor. 



In general the operator products, which appear on the right, are factors of 



WDk, 



which contain the factor D 3 , every weight of ojrarator product being represented once, 

 ciiid once only, from the weight 2 up to the weight of the single operator on the left- 

 hand side. 



It is important to remark that /o*i\ 



\* / 

 is a perfect partition* of the number 



2/fc+l ; 



because every lower number can be composed in exactly one way by the parts of the 

 partition. 



Art. 96. It will now appear that there exists an expression for 



D. 



corresponding to every perfect partition that can be constructed. 

 The general expression of a perfect partition is 



where a, /8, y, 8, ... are any positive integers, zero excluded. 

 The perfect partition ,~^ \ 



is the particular case 



= 1 , p = A\ y = 8 = . . . = 0. 



' Messenger,' 1890, p. 103. 



Eo 

 m 



