370 MAJOR P. A. MACMAHON ON COMBINATORIAL ANALYSIS. 



and note the result 



DA = 



and also 



where (<r } <r 2 cr s . . . <TI) is a partition of s and the sum is for all such partitions, 

 and for a particular partition is for all ways of operating upon the suffixes with the 

 parts of the partition. Thus 



DA/tA = ^A-s^A + M>-A + V'A-3 



If from the result of D, A v A*, . . . A A , we select the term pi-oduct 

 corresponding lattice will have as first row 



the sum of the numbers being .s, and if in the selected product we now operate with 



D, we can select a term product from the result, and the two minor operations may be 



indicated by the two-row lattice, 



the sum of the numbers T being t. 

 Hence if we take as operation 



and as function ^A/'AJ ^A, 



we will obtain a number of lattices of m rows and I columns, which possess the 

 property that the sums of the numbers in the successive rows are /t x , /* 2 , . . . \i m , and 

 in the successive columns X l5 X 2 , . . . X/, no restriction being placed upon the magni- 

 tude of the numbers. The number of such lattices is A, where 



and now transposition of lattices shows that 



yielding a proof of a law of symmetry discovered by the present author many years 

 ago. The process involves the actual formation of the things enumerated by the 

 number A. The secret of its success in this instance lies in the result D A /t, A,_ A . 



