THE FOUNDATIONS OF A NEW THEORY. 363 



Art. 2. With the immediate object of applying the method to the general case of 

 permutation, there being any number of identities of letters, we must first obtain 

 another solution of the foregoing problem. 



Let a ]( a. 2 , 3 ... be a number of quantities and a,, a z ,-a a , . . . their elementary 

 symmetric functions. Further, let 



j T\ d d d 



a, = D! = - h i -j -- h a T~ 

 rfoi 1 da 3 ' s rfa, 



and a, = ta^a.-, . . . 3 = (!') in partition notation. 

 We may take as operation and function 



D," and (1)", 



equivalent to (d/da } )" and a,*, which we had before, but more convenient as being 

 readily generalisable. 



Let D, = -c^', df denoting an operator of order s, obtained by symbolical multi- 



plication as in Taylor's theorem. Suppose the question be the enumeration of the 

 permutations of the quantities in a. l w> a. i w * . . . /-, where Sir = n. I say that the 

 operation and function are respectively 



D W1 D., . . . D,. and (1)". 

 Observe that this is merely the multinomial theorem for 



in partition notation ; and 



. . .)=!. 



Hence 



n! 



..D.D,, . . . D,.(l)- = 



IT, ! 7T, ! Wj ! . . . IT, ! 



the result we require. 



The important operator D, has been discussed by the author, t Its effect upon a 

 monomial symmetric function is to erase a part IT from the partition expression of 

 the function. 

 Thus 



. . .) = (p<r . . . ) = taffr .... 



* See HAMMOND, ' Proc. Lend. Math. Soc.,' vol. 13, p. 79 ; also ' Trans. Camb. Phil. Soc.,' /or. fit. 



t ' Messenger of Mathematics,' vol. 14, p. 164. ' American Journal of Mathematics,' "Third Memoir 

 on a New Theory of Symmetric Functions," vol. 13, p. 8 et seq., p. 34 ei seq. 'Trans. Camb. Phil. Soc.,' 

 vol. 16, part IV., p. 262. 



3 A 2 



