THEORY OF THE PARTITIONS OF NUMBERS. 97 



e, is the exponent of x due to the dividing lines when only the first . lines from the 

 left are retained, the letters to the right of the s ih line being arranged in alphabetical 

 order. If /A = (m l)(n-l) we know that e^ = / = mn/n. What is the relation 

 between e, and e^, ? To obtain e M _. we must clearly obliterate the last * Knes on the 

 right and arrange the affected letters in alphabetical order. Since the number of 

 letters is mn, if for e, we retain s lines which give 



we must for e^, reject s lines of power values 



mn ^1, mnp a , .. 

 Hence 



e *-> = e^smn+e, = e t 



and from the symmetry of the permutation we find also 



/,-. 

 so that 



an interesting result which foreshadows the theorem 



T _ V /IM (M-J.) T 



.LJ^-, U/ J-l,. 



The circumstance that the lattice function, when the lattice is complete, involves x 

 to the power ^mn(m 1) (n I), which is the greatest exponent of x that occurs in 

 the outer function, is consistent with the inner function being simply unity. 



Art. 21. I will now investigate an expression for L, ( o>, m, n). 



Suppose that a certain power of x arises therein from the conjunction ,,/, where 

 in S v > u, and let the Greek letter succession be 



where in the space A there is any suitable succession of letters in ascending order (of 

 subscripts) to v, and in the space B any suitable succession such that the subscripts 

 are in ascending order from u. 



The least power of x is obtained when in the space A there is the succession 



This gives the term C c''* ( "- 1) "-" +1 . 



The greatest power arises when in the space A is 



and this gives the term a; *" 1 "'. 



VOL. CCXI. A. 



