THE FOUNDATIONS OF A NEW THEORY. 



and, in the literal form, they are such that the product of letters in each of the 

 \ + \L -f- v rows and X 4- p + v columns is a x 6 M c', one letter appearing in each com- 

 partment of the lattice. This is the extension of the idea of the Latin Square which 

 was successfully considered in the former paper (loc. cit.), but now the enumeration is 

 given in a simpler form and from simpler considerations. 



In general, it has been established above that the Latin Squares based upon the 

 product 



are enumerated by the expression 



the simplicity of which leaves nothing to be desired. 



Art. 16. Consider the particular case of the general theorem which is such that 

 no compartment is empty ; the lattice has n columns and m rows. 



Pi 



= P 



" = 



the corresponding lattices being enumerated by 



These, when given the literal form, possess the property that the products of 

 letters in the successive rows are a p> b g 'c ri , a^fe'v 1 " 1 , . . . cf~b qm c T ~ respectively, and in the 

 successive columns a* 1 6" I c 1 ' 1 , a^&'V 1 , . . . cf'lf'c" respectively. 



Ex. gr. Suppose the row products to be a s 6, a 2 6 2 , a'fe 2 , a*, and the column products 

 a*, a 3 6, a 8 6, at 3 



D sl D| 2 D 40 (ro 4 )(To s "oT) 2 (To 01') 

 D 81 D| i (Io 8 )(10 > 6l) 2 (0?) 



D B 2(10) 2 (01)' 

 2. 



and the two lattices are 



