THE FOUNDATIONS OF A NEW THEORY. 



3G9 



and we can show that the numter A enumerates the lattices under investigation. 

 The operation D M| makes selections of every /i, of the / factors and erases a part 

 unity from each ; one minor operation of D Mi therefore is denoted by ft l units placed in 

 /A[ compartments of the first row of a lattice of / rows ; the operation D Mi adds on u 

 second row, in which units appear in fi 3 of the compartments, and so on we finally 

 arrive at a lattice possessing the desired property as regards rows, and as obviously 

 the column property obtains, the problem is solved. 



Ex. gr. Take X x = 3, A., = 2, X 3 = 1, /t, = 2, ,z 2 = 2, ^ = 1, /x, = 1 

 (1 s ) (I 2 ) (1) = a 3 a 2 a, = . . . + 8(2211) + 



The eight diagrams are 



and no others possess the desired property. 



We can now apply the method so as to be an instrument of reciprocation in 

 algebra. If we transpose the diagrams so as to read by rows as they formerly did 

 by columns, the effect is to interchange the set of numbers X ]( Xj, . . . X/ with the 

 set /ij, p, s , . . . p. m , and the number of diagrams is not altered. Hence the reciprocal 

 theorem. 



then (I 1 "))!** 1 ) . . . (1 M ") = . . . + A^jXj . . . X/) + 



a theorem known to algebraists as the Cayley-Betti Law of Symmetry in Symmetric 

 Functions. 



The easy intuitive nature of this proof of the theorem is very remarkable. 



Art. 5. In the above the magnitude of the numbers, appearing in the compartments 

 of the lattice, has been restricted so as not to exceed unity. This restriction may 

 be removed in the following manner. Consider the symmetric functions known as 

 the homogeneous product sums of the quantities a,, a 2 , otg, . . . viz.' 



/ h = (l), 



hi = (3) + (21) + (1), 



VOL. CXCIV. A. 



3 B 



