THE FOUNDATIONS OF A NEW THEORY. 373 



theory of Invariants), hut here the determination is representative as well as 

 enumerative, and has moreover analytical expression. 



Ex. gr. Take as function ( ( J87)(654)(321), and as operation D 13 D u Di 7 ; then 



b 13 D 15 D 17 (987)(654(321) = A, 

 where (987) (654) (321)= . . . + A (13.15.17) + .... 



One of the associated lattices is 



where observe that the numbers in the successive rows constitute partitions of the 

 numbers 13, 15, 17 respectively, whilst in the successive columns the numbers con- 

 stitute the partitions (987), (654), (321) respectively. The number of lattices 

 possessing this property, is A, and A is readily found to have the value 6. If we had 

 to find an expression for the number of row and column-magic squares of order 3, 

 it would be necessary to write down the sum of all products 



(762) (951) (843) 



formed from the first ( = n z ) numbers in such wise that the content of each 

 partition factor is 15 = n(n z +1), attention being paid to the order of the parti- 

 tions, and to take as operation DJ 5 or in general Dj^,,.,. ,>. The resulting lattices will 

 all be magic squares in which the diagonal property is not essential, and the result of 

 the operation upon the function will give the enumerating number. 



Art. 7. To resume ; in the lattice compartments we find invariably the numlxjrs 

 X,, /A I( v lt . . . Ag, /i 2 , 1/2, ... X,, p.,, v,, . . . such numbers being subject to certain 

 conditions for each row and each column. The assemblages of numbers in the 

 successive columns do not vary from lattice to lattice, but those in the successive 

 rows do vary from lattice to lattice. 



Let X, + pi -f-y, + . . . = A ; X ; + /A.J + >/ 2 -f . = /* ; 



A 3 + Ms + "3 + v, &c. 

 Then we have the following facts : 



(i.) The whole assemblage of numbers, A,, /*,, v lt . . . A.,, /u,.,, K,, . . . A 3 ^- A -v.^ . . . 



is unaltered from lattice to lattice, 

 (ii.) The numbers A, /*, v, . . . appertaining to the columns, and the numljers 



Pi q l r, . . . appertaining to the rows, are unaltered from lattice to lattice. 

 These conditions do not define the lattices in question, localise other lattices 

 comply with them, viz., those in which, the whole assemblages of compartment 

 numbers remaining unchanged, the column partitions, while satisfying the condition 

 (ii.), are other than 



