MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. 88 



so that, substituting, 



hence 



Art. 28. Further, summing each side with respect to m, 

 SSSN(,a a ...e/ n ...) 



,n n" 



n! 



but the sinister is of the form 



SN (w'.Wj) (see Art, 20) 

 and thus has the value ! ; hence 



an interesting result. 



Art. 29. From a previous result 



hence 



2N(a,rtj...a m +l) = 2N 



n n 



and it may be observed that the miml>er8, included in 



are the conjugates of those included in 

 Art. 30. Also since 



,a a ...o M +l) = N m ,,i'-n-SN(a 1 cf,...o m _,l), 



n a 



2N (a I a i ...a.-, 



a 



and this leads to the relation 



E.g., for p = 3, m = 2, p-ni+2 = 3, 



3N 3 , 1 ,+2N,, 1 , = N,.,. 

 verified by 



8.1 + 2.4 = 11. 



M 2 



