THE FOUNDATIONS OF A NEW THEORY. 379 



the terms written comprising all monomial functions whose partitions contain exactly 

 s parts ; and 



(O x ) (0") (0") ...=... 4 B,(0') -|- . , . 



If P fiPt ... p. denote the number of permutations of the numbers p^p^, . . . p., 

 I say that the above lattice theory establishes the relation 



"i "fft . . .p. A?,?, . . . p. T "p,'p,' . . .p,' A-p t 'p,' .../>.'"!" 



Ex. gr. Observe the two results 



(!)(!) (1) = (31) + 2(2") + 5(21*) + 12(1*), 

 (0 8 )(0)(0) = 4(0 2 ) + 15(0 8 ) 4- 12(0*), 



and verify that the relation given obtains between the coefficients. 



Art. 11. This zero theory is really nothing more than a calculus of binomial 

 coefficients, which enables the study of their properties by means of the powerful 

 instruments appertaining to the Theory of Symmetric Functions. The Law of Sym- 

 metry established by the author in the Second Memoir (loc. cit.) is easily established 

 by means of the lattice ; it may be stated in a simple form, because all the functions 

 (0),(0 2 ),(0 8 ), . . . are to be regarded as having the same specification, viz. (0); further, 

 the specification of (O x )(0")(0') ... to s factors is (0*)). Select all the products 

 formed from a given number of zeros which have the same specification (0'), and 

 attach to each a coefficient equal to the number of permutations of which it is sus- 

 ceptible. Denote the sum of such products by SCo (0*)(0 ( ')(0 1 ') .... Similarly, 

 for a specification (0*) denote the sum of such products by Co (O p )(0'')(0 r ) .... 

 Then 



SCo(O x ) (0") (0") ... = ...+ A(0>) + . . . 



2Co(0')(0)((K)_. ..=... 4- A(O-) + ... 



the coefficient A being the same in both cases. 

 Ex. gr. Verify that 



2(o 3 )(o) + (o 2 ) 3 = . . . 4- i2(o 3 ) + ... 



3(0*) (O) 2 = . . . 4- 12(0 2 ) 4- ... 



Art. 12. I continue the general plan of this paper, viz. : I do not attempt the 

 solution of any particular problems, unless they are suggested by the general course 

 of the investigation, but rather start with definite operations and functions, and 

 seek to discover the problems of which they furnish the solution. This is the 

 reverse process to that employed in the ' Trans. Camb. Phil. Soc.' (loc. cit.), where I 

 particularly investigated a number of questions more or less directly associated with 



3 c 2 



