THEORY OF THE PARTITIONS OF NUMBERS. 349 



and also 



GF (I i a") -*~GF (/- !;") = GF (/ ; a- 1 , a- 1). 



The formation of the relation in any particular case presents no difficulty. When 

 the lattice has k singly detachable nodes, the right-hand side of the relation involves 

 2*-l terms. 



Art. 5. When the part magnitude is unrestricted, or I = oo, the equations become 



(a+b+c)GF(;a,6,c) 



= GF(oo;a-l,&,c)+GF(oo;a, 6-l,c)+GF(oo;a, 6,c-l) 

 -GF(oo;a-l ) 6-l,c)-GF(oo;a-l,6 ) c-l)-GF(oo;a,6-l,c-l) 

 + GF(oo;a-l,6-l,c-l) 



(.5a)GF( co ;,, .... a.) = {i_(i_^)(i_,,,)... (i-p,,)}GF(oo;a 1 , a,,..., a.) 



and the modified forms are easily written down. 



. Art. 6. The next step is to deduce the corresponding relations between lattice 



functions. From the relation 



GF( oo ; 1)03 , ...,.) = L( OB ;,....) 



we find 



L ( . ; , 6, ,) = 



L(oo;a,) = L(oo;a, -l); 



L(oo;a, b) = L(oo;-l 

 L ( oo ; a, a, a) = L ( oo ; a, a, a 1 ) ; 



L ( oo ; a, a, 6) = L ( oo ; a, a- 1, 6)+ L ( oo ; a, a, b- l)-(2a+b 1) L ( oo ; a, a- 1 , b- 1) ; 

 L(oo;tt,6,6) = L(co;a-l,6, &)+L( oc ; a , M-l)-(a+2b-l) L( oo ; a-1, M-l); 

 L(oo;a, 6, r) = L(oo;a-l, b, <) + L( oo ; a, &-1, r)+L( oo ; a, 6, c-1) 



+ L(oo ;a , 6-1, c- 

 + (a+b+c-2) (a+b-f c-1) L( oo ; -i, 6-1, c- 1). 



In the case of n unequal rows, if we write symbolically, 



^.L(oo; o,,o,, ...,a,) = L(oo; a,, a,, ...,a,-l, ..., 

 -1) ... (JTa-m+1) = X" ; 



then 



