MAJOR P. A. M\. \i.\irON ON THE COMPOSITIONS OF NUMBERS. 97 



From the relation 



we obtain 



9(abcd) = M(a)(&c)(rf)}-M( + &, c)(d)}-0,{(a)(b, c+rf)}+^{(a + 6, c+d)} ; 

 and there is no necessity to give further examples. 



SECTION 5. 



Art. 49. The differential operator, of order s, that is so frequently of use in the theory 

 of symmetric functions, viz. : 



can now be employed. 



Remembering that operating upon monomial symmetric functions, 



D a (a)=l, 



D a (f>) = unless b = a, 



D a DJ) e ...(a&c...) = 1; 



and generally that D a obliterates a number a from the partition of a function and 

 causes it to vanish if no such number presents itself, it is clear that 



a A t V '* D ' D > D < 



and thence if we write 



according to a law derivable from that which defines 



() (&)(e)...} (see Art. 38), 

 we find 



Art. 50. Observe that in the paper to which reference has been made it was shown 

 that 



Two consequences flow from this fact. 

 Firstl 



which is a theorem of reciprocity for the numbers 



N (...). 



VOL. CCVII. A. O 



