MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS. 101 



SECTION 6. 

 The Symmetric Functions A ftftft , (t flflfl ... . 



Art. 54. These two new functions are of fundamental importance, not only in this 

 investigation, but in the theory of symmetric functions generally. 

 In regard to the algebraic equation 



AI, A,, l> n , ... are the homogeneous product sums of the roots and the two sets of 

 elements 



have reciprocal properties which it is useful to briefly glance at. 



We have 



/,, = , = (!), 



A3 = a l 3 -2a 1 a 3 +a a = (3) + (21) + (I 3 ), 

 and, in general, 



The two series of elements are connected in such wise that, in any relation between 

 the elements, the symbols a, k may be interchanged. Thus, from 



a, a -3a a = -2/h 3 



is derived 



A, 3A 3 = 2 1 - 



As a particular case it is found that a, is the same function of the elements 

 hi, AS, /J 3 , ... that h, is of the elements a,, a a , 3 , 

 If functions of the elements /i u A,, fi 3 , ... be denoted by 



we see that, if 



then 



showing that 



is an absolute invariant qud the transformation which replaces the elements 



AI, A,, A,, ... 



by the elements 



J 



