MA. TOR P. A. ifACMAHON ON THE COMPOSITIONS OF NUMBERS. 



113 



SECTION 8. 



Art. 72. I recall that the number of ways of distributing numbers (or objects) 

 specified by (n Wt v''D w> ) 



into m different parcels, is given by the series 



\Pi)\frJ\P 



/m\ (m +p l 2\'' fm +p t 2\'' (m + p a 2\* 1 



~uA PI ) ( P, ) \ p* 



. fm\ fm + />, - 3 V" fm + ,- 3\'- fm + 8 - 3 V 1 

 h \2/\ *./('*)(*) '" 



and this, for brevity, I write 



v 



* n , 



Let 



1 

 \ * / 



denote the number of distributions, associated with a descending specification 

 containing exactly m parts, and write this 



N., 



when there is no risk of misunderstanding. 



Following the proof of Art. 23, it may be proved that 



/n-rn+r 



and nlso 



N = F -I 



A1 m * m I 



and thence 



XT =G ^ tt "^MG +(^"MG +( } m * l/ """" A \G 



Art. 73. From this relation the following results are obtained : 



n = 3. 



VOL. COVII. A. 



