130 MAJOK P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS, 



but no essentially new relations are thus obtainable ; for take a simple case 



leading to 



true for all valiies of m. 



Ifwetake 

 we are led to 



2 (N M -N m _ 1 ) 1 c, 8 c, + i = (N.- 

 and if the former relation be written 



the latter is merely f(m )_ f(m _ l} = . 



and further multiplication by b leads to the series of which the general term is 



so that no new information is obtained. 



Art. 114. The operator relation of the form D,D t = a linear function of 



7) 

 u 



is not difficult to obtain. 

 I find that 



and thence the formula for 



D,D,D U 



follows by taking D B as the operand on each side and then reducing the products 



D,D M , D, +1 D,,, D. +3 D B) ... 

 by the formula for D,D e . 



I find that 



-(' 



U \ /t\ s+i\ f u \ 



'+ 



