B60 



MAJOR P. A. MACMAHON: MEMOIR ON THE 



ln general, the determinant numerator of GF ( oo ; ,, . . , *.) involves x explicitly 

 as exhibited in the determinant 



7%e Restriction on the Part Magnitude. 



Art. 21. I pass on to consider the case in which the part magnitude is restricted by 



the integer I. 



I take as my point of departure the functional equation 



(\- PI ) (1 -p 2 ) ... (1 -p.) GF (I ; a,, o . . .', a.) = x s GF (I- 1 ; a ls o,, . . ., a.), 

 and, by means of the relation 



. Ll (t ', Qi\, Cl2, . . . , dnj 



convert it into a functional equation for the lattice function. 

 For the orders 2 and 3 we have 



= (a+b) {L (I o-l, 6)+L (/ ; a, fc-l)}- 



a+b) L (Z ; o-l, 6-1) ; 



= (a+b+c) {L(Z ; o-l, 6, c) + L(Z ; a, fc-1, c) + L ( ; a, 6, c-1)} 



(+b+c-2)(a+b+c-l) (a+b+ c) L(l ; o-l, 6-1, c-1) ; 



