THK FOUNDATIONS i>F A NK\V 



371 



Ex. gr. We have 



i.e. 



= 18 



18(2211) -f . . . 

 = . . . + 18(321) + . . . 



.UK! we must have 18 lattices; now eight of these, in which the compartment 

 numbers do not exceed unity, have been depicted above ; the remaining 10 are 



Art. 0. The next problem I will consider is that in which the magnitude of the 

 compartment numbers has a superior limit k. 



Let k, denote the homogeneous product sum of order s in which none of the 

 quantities a 1( a. 2 , 3 , . . . is raised to a higher power than k. Ex. gr. If k = 2, & 3 

 will be (21) + (1 s ), and not (3) + (21) + (I 3 ). 



We have 

 and 



Take as operation 



, = 



where A = k, 

 if A > k. 



and as function k^, k^, . . . (\, and we will obtain a number of lattices of HI rows and 

 I columns which possess the property that the sums of the numbers in the successive 

 rows are /x x , fa, . . . p,,,, and in the successive columns Xj, A.,,, . . . A/, the magnitude 

 of the compartment numbers being restricted not to exceed /. 

 The number of such lattices is 



D,,D M1 . . . !>. * Al * A . . . . /,- = A, 



, . . . fi m ) -f . . . , 



where k^ ...** 



and by transposing the lattices 



1 A, . . . A/) 



establishing a law of symmetry in symmetrical algebra. 



3 B 2 



