388 MAJOR P. A. M.M'MAIION: MKMOIR ON THE 



It is a remarkable fact that this elegant result appears to be valid whatsoever the 

 equalities may be that present themselves between the integers a,b,c,d ..... The 

 discussion of this and the interpretation of GF (I; a, b, c,d, ...) when a, b, c,d, ..., 

 are not in descending order must be deferred to another occasion. 



I add the general result for equal rows which has now been established 



GF (I ; a") 



(H-n)...(l+a+n-l) (l+n-1) ... (l+a+n-2) (l+l) ... (l+a) 



(n) ... (a+n-1) (n-1) ... (a+n-2) (1) ... (a) 



Art. 30. I recall that, in Part V., the formula was given 

 GF (I a, b, c, d, ...) = LoGF (I SaJ + I^GF^-l ; 2a) + ... + L.GF (l-p 



wherein L, is the sub-lattice function of order s derived from the lattice whose 

 specification is (abed...). So far it has not been feasible to directly determine an 

 expression for L,, but as we now know the expression for GF (I ; a,b,c,d,...) it is 

 possible to find expressions for Lj, L 2 , L 3 , ..., by giving I the values 1, 2, 3, ..., in 

 succession in the resulting identity. One interesting result was, however, directly 

 determined, viz. : 



L,(co ; a, b) 



but I do not give the proof of it at present, as the subject of the sub-lattice functions 

 has not yet been worked out, and they are, in fact, no longer necessary for this part 

 of the general investigation. 



Art. 31. Valuable information, concerning line or one-dimensional partitions, is 

 furnished by putting / = 1 in the general formula. 



The partitions that are then enumerated are those in which every part is unity, 

 there being not more than a, units in the s th row; if thence we proceed to line 

 partitions by adding the units that appear in each row we clearly get a system of 

 line partitions such that the s th part is limited in magnitude by the integer a. ; or 

 the system comprises all partitions contained in or subordinate to (a lt 2 , ..., ), viz., 

 such that the first part > a lt the second > 2) ..., the n th > . 



Denoting the generating function of these by 



, a 2 , ...,; n), 



X 



