102 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OP NUMBERS. 



Art. 55. With these necessary preliminary remarks I define a new function of 

 weight 71, viz. : , 



where pip a pa--- is any composition of the number n ; of the given weight there are 



such functions, one of which is clearly 



The complete definition is given by the multiplication law 



= h +/i 



where the functions , , 



are, or are not, of the same weight. 

 Art. 56. A second new function 



is similarly defined by the same law ; viz., 



What follows applies generally to both of the new functions. 



Art. 57. Since the multiplication is commutative, we have the first important 

 property, viz., 



PiPi P-9t9ifi '"Pi Pi- \,P-+9i, 



~ "' ... i- - >"' 



p- 



Art. 58. Every product of elementary functions is expressible in terms of the new 

 functions, e.a., 



and in gene ra l 



where, in h { }, the sum of the coefficients is 



2- 1 . 



These relations show that , , 



fi pip*~p- ~ "p.-. 



Art. 51). Similarly 



- 



and in general 



It, moreover, we define 



2 ...q t ) (r } r 2 ...r u )} 



