THEORY OK THK PARTITIONS OK NUMBERS. 371 



Siniil.-.rly, from the ( Jreek -letter successions \\hidi involve / nted letters, I derive 

 the sub-lattice function of order t, and thence, by previous reasoning, arrive at the 

 ^em-rut ing function when the part magnitude is restricted by the integer I, 



I. 22 . 22) = SL<( ; 22 ; 22)(l-t-|-l)(l-t-l-2) ... (l-t+8) 



Art. 36. The method is generally applicable to any incomplete lattice in three 

 dimensions. I work out in detail the case in which the points of the lattice form the 

 8 summits of a cube, in order to show that the result obtained, in Part II., Section 7, 

 in quite a different manner, is verified. That result, with modified notation, was 



Generating Function 



where 



P(*) 



R(a;) = 2z IO +2z ll +3x I> +2.c 18 +2a; 14 . 

 I shall now show that 



are, in fact, the sub-lattice functions of orders to 4 which appertain to the lattice 

 formed by the summits of a cube. 



I write down the 48 permutations of the Greek letters and over each the arrange- 

 ment of the first 8 integers from which it is derived, the lower layer of numbers 

 being placed to the left : 



3 B 2 



