THE FOUNDATIONS OF A NEW THEOEY. 367 



again D/D! (1 s ) (I 2 ) = D 2 Dj (1 2 )(1) and the two operations together give us 



= D, 0) = 



1 and the complete lattice representation is 



which is none other than the graph of the partition (32 2 1) or of (431) according as it 

 is read by rows or by columns. We might also have operated with D 4 D 3 D, upon 

 (1 s ) (I 2 ) 2 (1) and, in general, if (TT^TT^ . . .), (p } p<>p 3 . . .) be conjugate partitions, 

 we obtain their graphs either by operating with D^D^D^ . . . upon (P 1 ) (1 P )(1 P1 ) . . . 

 or with DJ^D,, . . . upon (l")(l") (!") 



Art. 4. I proceed to consider some less obvious but equally interesting examples 

 of the method. The diagrams obtained depend upon the law by which the operation 

 is performed upon the function which is the operand. The operator D. in connexion 

 with symmetric function operands is of commanding importance. It would be 

 difficult to imagine an operation better adapted to research in combinatorial 

 analysis. We shall find later that an analogous operation exists which can be 

 employed when symmetric functions of several systems of quantities are taken 

 as operands. As an example of diagram formation, take as operator D 4 D a 2 and as 

 function (3) (21) (2) (1) (l) the weight of operator and of function being the same. 



We have 



the eight terms arising from the partitions 31, 22, 211 of the number 4. The dots 

 take the place of the picked out partitions. 



Hence [ ' D 4 (3)(21)(2)(l)(l) 



= (2)(1) + 2(21) (2) (1) + (3) (I) 3 + 3(3) (2) (1) + (3) (21). 



The operation D 4 breaks up here into eight minor operations ; taking any one of 



