92 MAJOR P. A. MACMAHON: MEMOIR ON THE 



As a rule, with some exceptions, the inner lattice function changes in passing from 



a lattice to its conjugate. 

 Thus it will be found that 



(22221) and (54) being conjugate partitions. 



Exceptionally, if it be proved that the inner lattice function of a complete lattice 

 is unity, the function obviously does not change on passing to the conjugate lattice. 



Another exception appears to be 



IL ( oo ; ml") = IL ( oo ; + 1 . I"' 1 ), 



ml" and >t + l . I"" 1 being conjugate partitions and there may be others. 



Art. 16. My next object is to obtain the lattice function for / = co which 

 appertains to a lattice of two unequal rows and to find the form of the inner lattice 

 function. 



The first step is to establish the relation 



L( oo ;/,) = L( oo ; , &-l)+af + *-'L( oo ; a-1, &-1) 



Consider the Greek letter succession a"/? 6 , where a ^ b. 



The whole of the permutations derived from the lattice terminate in one of the 

 following ways 



/8;/8|; /8|a a ; ... j8|a-*, 



since a cannot occur more than a b times at the end of the permutation by reason of 

 the fundamental property of a permutation. Permutations which terminate in the 

 manner j a' where * > clearly give rise to a factor x? +b ~' in the associated powers 

 of x \ the other factor will be due to all of the permutations of the succession a"~ s /3* 

 it-Inch terminate ivith /3 ; that is to say, the other factor will be 



L(oo ; a s, 61). 

 Hence 



L( oo ; ab) = L( oo ; a, 6-l)+s*-'L( oo ; a -, &-1), 



as was to 1* shown. 



Now assume the truth of the relation 



