126 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 







Art. 103. Again, in the same formula, put 



r, = n-2, c 2 = 0, r 3 = r., = ... = 0, 



Wefind 2N.. a .- = N^ + fN.-N,.-,)."-!. 



We obtain, from this, a useful result by writing 



n m + \ for m, 

 for then 2N.-. +I ,-2 = N B _ m+1 . 1 + (N B _ m+1 -N,_ m V.-i. 



Art. 104. Observe that XT -,. T 



N m ,i = JN n _ m+lil , 



so that by addition and subtraction we obtain 



m+li21 '.-2 = N m-1 



or, as we may conveniently write these relations, 



(N.+N.-.+On = N., : 

 (N m -N B _ m+1 )2i"- 2 = (N 7n -N B _ B1+1 )i'- 1 : 

 = (N.-N.-O,-!. 



These are the relations connecting N m and N,_ m+ i qud the subscript 21"~ a analogous 

 to those connecting the same symbols qud the subscript 1". 



Art. 105. From any operator relation we can immediately derive a relation between 



the numbers N m by substituting for 



b'VV... 

 the expressions 



and this it is convenient to denote by 



N (<r) m , ,-,/.... 

 Art. 106. Thus, corresponding to the operator relation 



6D 3 = V l (D l 

 we obtain 



As a particular case put 



c, = n-3, r z = r 3 = c 4 = ... = 0, 

 so that 



