THE FOUNDATIONS OF A NEW THEORY. 375 



connected with D, 3 (2) 2 (l) 2 = G, 

 and (2) 2 (1 8 )= . . . + 2(2*1 2 ) + . . . 



connected with D, 2 D, 2 (2) 2 (1 2 ) == 2. 



In the first case the row and column specifications are 2, 2, 2, and 2, 2, 1, 1, respec- 

 tively ; and in the second ca.se 2, 2, 1, I, and 2, 2, 2, respectively. 



The first case yields the six lattices 



the second case the two lattices 



If we transpose the six lattices we ohtain four lattices in addition to these two, 

 viz. : 



The first pair of these would be derived from D 2 2 D, 2 (2)(1 2 )(2), and the second 

 pair from D S 2 D 1 2 (1 2 )(2) 2 . Hence it is clear that to ohtain identity of enumeration we 

 must multiply (2) 2 ( I 2 ) by a number equal to the number of ways of permuting the 



(2 + 1) ' 

 factors which have the same specification, viz., by jTrTj = 3- 



2 ! 2 ! 



The corresponding multiplier of (2) 2 (1 2 ) is 9 i -^~, = 1. 

 Let then an operand be 



L L , L^, L 3 , . . . denoting different partitions of the same weight, 

 M 1( M 2 , M 3 . . . 



&c., &c., Ac. 



we attach a coefficient 



Ill/; !/,!... 



! m, ! OTJ ! . . . ' ' * 





