THEORY OF THE PARTITIONS OF NUMBERS. 105 



Assume that GF (I, m,n) = J (I, m, n) when I does not exceed mn ; then putting 

 I = mn + 1 in our equations, we find that 



GF(mn+l, m, n) = J(wn+ 1, m, n) ; 

 and thence by induction 



GF (I, m, n) = J (/, m, n) when I > mn. 



Art. 28. I now write the fundamental relation 



in the form 



GF(l,*m*)5 



where 



and assume that GF (I, m, n) is a product of powers of factors of the two types 

 1 x' + ', 1 ce*, or (l+s), (s), where the powers may be positive or negative integers. 

 I thus write 



I -HCl+i). !!,<), 



leading to 



and 



Putting herein I = 0, I' = I, 



and on the assumption as to form the main theorem is established. 

 VOL. ccxi. A. f 



