346 MAJOR P. A. MxcMAHON: MEMOIR ON THE 



formation and solution of certain functional equations which lead in the first place 

 to the required generating functions, and in the second place to an exhibition of the 

 forms of the sub-lattice functions. To previous definitions I here add the definition 

 of the inner lattice function when there is a restriction upon the part magnitude, and 

 it will be shown that the generating, lattice, and inner lattice functions satisfy 

 certain functional equations both when there is not and when there is a restriction 

 upon the part magnitude. 



There are two methods of investigation available. We may commence with a 

 study of the Greek-letter successions (Art. 6, et seq., loc. cit.) from which the lattice 

 functions are derived, and having obtained the functional equations which they 

 satisfy, proceed thence to those satisfied by the generating and inner lattice functions ; 

 or we may reverse the process, and, by a prior determination of the equations apper 

 taining to the generating functions, arrive at those satisfied by the lattice and inner 

 lattice functions. 



Both methods have been of service. 



The results, herein achieved, are complete so far as the lattice of unequal rows and 

 the particular question under consideration are concerned. They are elegant and 

 algebraically interesting. In proof of this, it will suffice to say that the generating 

 function is unaltered when the lattice is changed into its conjugate. The subject 

 thus swarms with algebraical relations which are established intuitively. 



Other results are obtained of a more general and extensive character which mark 

 out the path of further investigation. 



Art. 1. I recall that for the lattice of two unequal rows, containing a, b nodes 

 respectively, the established results are 



Inner lattice function = IL ( <x ; a, b) = l+st?'* 1 ( a ~"b) . 



Lattice function = L ( oo ; M = (1) (2)...(a+b) f 6 



Generating function = GF ( oo ; a, b) = - 



We have yet to determine IL(l;a,b), L (/ ; a, b), GF (Z ; a, 6), where an inner 

 funct.on, for a restricted part magnitude, is defined by the relat-ion 



U+n)...(H-aH-n-l) . 



