MAJOR P. A. M\< \I.\HON ON THE COMPOSITIONS OF NUMBERS. 125 







and then (*+)D()D = ()D(+")D = (OD( + )D I 



or, as these relations may be written, 



d.+t d u = d, d t +* = d t d u +, ; 

 (s+t, tt) D = (*, *+) = (, u+*) D , 



with the usual multiplier (viz., 2), if either 



x 4- / = '*, or f + = ., or + = . 



Art. 101. We are led to the series 



(81)0-2(2%, 



(41) D = (32) D) 

 (51) D = (42) D = 2(3') D , 

 (61) D = (52) D = (43) D , 

 (71) D = (fi2) D = (53) D = 2 (4 a ) D) Ac., 

 and generally if (^/'...), (^,-...) 



be functions of the same weight and degree, viz., 



Application of the Foregoing to the Genwatiiig Function. 

 Art. 102. It has been established that 



We will first of all examine the result of the equivalence of operators 



2!D, = D 1 '+(1-X)D, 

 (see Art. 93 qud the operand on the right-hand side). Write the operand 



m = 3, 



2N 3 ,, = N 8 . 



verified (from the tables) by 



2.48 = 93 + (l5 



