370 



MAJOR P. A. MACMAHON: MKMolR ON Till. 



dimensional partition whose successive parts specify the successive rows of the lattice, 

 so an incomplete lattice in three dimensions is defined by a two-dimensional partition 

 whose successive rows specify the successive layers of the lattice. 



I shall suppose these layers to be in or parallel to the plane of xy which is the 

 plane of the paper, and the axis of z to be perpendicular to the plane of the paper. 

 Descending order of magnitude of parts placed at the points of the lattice is to be 

 in evidence in the three directions Ox, Oy, Oz. 



Art. 34. Consider the simplest case of a complete lattice, the points forming the 

 summits of a cube. The two-dimensional lattices a^/Si/?,, a 2 a 2 ;8 2 /J 2 , in and parallel 

 to the plane of the paper are superposed to form the three-dimensional lattice. 



Suppose that the first 8 integers are placed at the 

 x points of the lattice so that descending order of 

 magnitude is in evidence in the directions Ox, Oy, Oz, 

 e.y., one of 48 such arrangements is as shown. 



I associate with the first and second rows of the first 

 layer the letters a,, /8i respectively, and with the first 

 and second rows of the second layer the letters a , /8 2 

 respectively, and then from the illustrated arrange- 

 ment of the first 8 numbers I derive a Greek-letter 



succession in the following manner : I take the numbers in descending order of 

 magnitude and write down the Greek letter with which the position of each number 

 is associated : thus the arrangement above gives 



87654321 



Art. 35. In this Greek-letter succession we have to note 



(i.) A /3 which is succeeded by an a, 



(ii.) An a which is succeeded by an a with a smaller suffix, 

 (iii.) A /8 which is succeeded by a /3 with a smaller suffix. 



If a letter which is thus noted is the s th letter in the permutation I associate with 

 the permutation the power x* 1 , and taking the sum of these powers in respect of the 

 whole of the permutations associated with and derived from the lattice I obtain the 

 lattice function 



2.x 2 '; 



and, following the reasoning of Part V., Art. 6, I derive the generating function for 

 partitions at the points of the lattice, the part magnitude being unrestricted, viz., 



