348 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



Hence the functional equation 



GF (/ ; a, b, c) -^ + "GF (I- \;a,b, c) 



= GF(l;a-l,b,c)+GF(l;a,b-l,c)+GF(l;a,b,c-l) 



In the general case of n unequal rows we have the theorem for GF (I ; a lt a,, . . . , a n ) ; 

 for if p, be a symbol such that 



p.GF^jaj.Oj a.) = GF(J;a 1 ,a ...,a.-l, ...,), 



it is readily seen that 



(l-p,)(l-p a )...(l-p,,)GF(/; a,, ob, ...,a.) = a*GF(Z-l ; a,, a,, .. .,). 



This equation is, at first sight, only true when there are no equalities between 

 the numbers a,, a a , ..., ; but in the sequel, when an algebraic expression of 

 GF(J; <*!, a,, ..., a.) has been found, it will be seen to be true universally as an 

 algebraical identity. 



Art. 4. However, the formula may be modified, in the direction of simplification, 

 when the rows are not all unequal. 



For a given lattice we require to know how many nodes may be singly detached 

 and yet leave a contained lattice. Thus in the three-row lattice illustrated above it 

 is clear, the rows presenting no equalities, that we may detach singly either of the 

 nodes lettered A, B, C ; but in the case now given 



. A a nodes 

 b nodes 

 b nodes, 



it is seen that we can detach either A or C only, so that the resulting functional 

 equation is 



GF (/ ; a, b, b) -a- + GF (I- 1 ; a, 6, 6) 



= GF (/ ; a-1, b, b)+GF (I ; a, b, b-l) -GF (I ; a-l, b, 6-1), 



which is to be compared with the equation appertaining to a lattice of two unequal 



rows 



Similarly we derive the equations 



GF (I ; a, a, b) -z*- + GF (I- 1; a, a, b) 



= GF(l;a,a-l,b)+GF(l;a,a,b-l)-GF(l;a,a-l,b-l), 

 GF(l;a, a, a)-a*GF(J-l ; a, a, a) = GF(/; a, a, a-l) ; 



