86 MAJOR P. A. MxcMAHON: MEMOIR ON THE 



function turns out to be simply unity ; a principal object of this investigation is to 

 establish that for the complete lattice the inner function is invariably unity. This is 

 consistent with the result conjectured in Art. 2. 



In regard to incomplete lattices the inner function is unity in special cases. The 

 determination of its form for the general incomplete lattice is apparently a very 

 difficult matter, which is reserved for future consideration. Its actual form for the 

 lattice of two unequal rows will be determined presently. 



Art. 11. There is also a vitally essential representation of the lattice function 

 as a sum of sub-lattice functions, which forms a natural bridge from the function 

 GF ( oo ; Oj, 03, ...) to the general function GF (1; a lt a 3 , a 3 , ...). When the lattice 

 function was formed from the permutations of the Greek letters, every permutation 

 had s dividing lines where s ranged from zero up to a maximum value p., which has 

 not yet been determined. That portion of the lattice function which is derived from 

 those permutations which involve precisely s dividing lines I name the sub-lattice 

 function of order * and write it 



L, ( oo ; a^aj ...), or L, ( oo ; m, ri), or simply L,, 

 if no confusion arises from the abbreviation. 

 """ L-fc, 







In the elementary examples already dealt with 



L ( oo, 2, 2) = L ( co ; 2, 2) + L, ( oo ; 2, 2), 

 1 + x* 



L ( co, 2, 3) = Lo ( oo ; 2, 3) + L, ( oo ; 2, 3) + L 2 ( oo ; 2, 3), 



1 + x*+x s +x* + x* 



L(oo;3,2,l) = L +L, +L 2 + L 3 , 



= 1 +x>+2x 3 +2x*+2x* 



It will be observed that LO is invariably unity. 



Art 12. In terms of these sub-lattice functions I now define the new and more 

 ;ice function L (1; ,, a,, 3 , ...), in which I replaces oo. I write 



and also a general sub-lattice function 

 L .(^-a 1) a 1 ,a 3> ...) = (l_j 



oo 



