THEORY OF THE PARTITIONS OF NUMBERS. 101 



I observe at this point that the substitution of I m n for / converts the factor 

 (l+l) (1+2) ... (l+m+n-1) into 



so that it is unchanged except as to sign and a power of x prd,i. 



Art. 24. Some particular cases of the fundamental relation are instructive. 

 Thus 



j 2 2) - d+l)...(l+4) + (l)...(H-8) a r' 

 (1)...(4) 



. 



wl 



(l)(2) a (3) 



Observe that 



= (l + l)(l+2) a (l+3)L(oo,2,2), 



showing that L ( oo, 2, 2) is a factor of the numerator. 



It appears that in general L( oo, m, n) is a factor of the numerator. 

 Thus 



GFH 3 3^ - (l+D d+2) a (1+3) 3 (l+4) a (1+5) L ( .. 3. 8) 



(1)(2)...(9) 

 and since 



L f, 3 PI -WW (8) (9) 

 (2)(3) 2 (4) 



. ox q+3) 3 (l+4) a (1+5) 



(l)(2) 2 (3) 3 (4) a (5) 



= |LM|,!LM|,|LM|3 when m = 3. 



Art. 25. I have arrived at the expression for GF(/, 2, n) in the following manner. 

 We have 



and 1 determine L,( oo, 2, n) from the Greek letter succession ; for suppose s = 3 and 

 a succession to be 



where ^,, j9 a , J9 3) p 4 ; 7,, 9,, ? 4 may have all integer (including zero) values subject 

 to the conditions 



A proper permutation with three dividing lines is thus secured, and we have to 

 perform the summation 



which is the expression of L 3 ( oo, 2, n). 



