FUNCTIONS IN THE THEORY OF I.\TK(!I;.\1. KQUATIONS. 107 



Since the product 



is convergent, we may choose a positive integer, w,, greater than n, which is such 

 that 



and therefore that 



for all negative values of X. 



Also it is clear that we may choose a negative number A, whose numerical value is 

 so great that 



'nn- 



for X s A. Hence, for these values of X, we have 



It follows from (14) that 



and therefore from (13) that, as X tends to oo, P (X) tends to a finite limit different 

 from zero. 



Since P (X) and D (X)/A (X) both tend to finite limits different from zero, as X tends 

 to oo, it is obvious from (12) that 



M + M 1 / \ 



n , ( l -E 



lim -" = ' V A " 



must be finite and different from zero. As this can only be the case when the number 

 of factors in the numerator is equal to the number in the denominator, it is evident 

 that m = 1.* Hence from (9) we have 



^+I = T.' ......... ... (15) 



As a corollary we deduce from (8) the inequality 



which was employed in III., 17. This is primarily true for n greater than fi, but, by 

 a suitable choice of v, it is evidently valid for all values of n. 



* Previous investigators seem to have overlooked the fact that the value of m is not obvious. They 

 have all tacitly assumed m = 0. 



