940 



where s is some integral number, all the coefficients a,n(x) are zero 

 from and after the value m =^ 2^ and thus a^ is also zero at those 

 points. But we can always find among these points such as satisfy 

 the condition that, corresponding to a number Q, chosen arbitrarily 

 great, there exists a value of 77* ^ Q for which the quantity 



I 

 |a'"(.t;)|'« (99) 



is more than a certain fixed amount x greater than the limit rz^ of 

 that quantity, which is zero: we need only choose the number s 

 greater than Q and m = 2^-^, and then a,n {.c) is equal to — 2 in the 

 points corresponding to those .s*-valiies, so that the quantity (99) has 

 a vahie which is greater than unity. Thus we cannot assign a 

 number m, independent of x such that in the loltole domain («) 



Wm (.^;)i < {ax + f)'" , for m > m, , 

 and this was the very uniformity-supposition of N". 4. The supposition 

 under A however is satisfied, because | a;" — 1 | is at most equal to 

 2 for points of the circular domain of radius unity and thus less 

 than (1 -|- e)'", that is less than 



[a (1) + fj'" 

 where 8 may be prescribed arbitrarily small, if only m be chosen 

 large enough. Again the supposition B is satisfied: in the first place 

 we immediately see that at all points of a circle arbitrarily little 

 smaller than that of radius unity the quantity fix is equal to 1 so 

 that there are in an arbitrarily small neighbourhood of the circum- 

 ference of the latter circle points where a^ is equal to the upper 

 limit of that quantity for the closed domain of that circle: this, 

 though not exactly the same as the supposition B, agrees with it 

 as to its consequences, viz. the validity of formula (7), and it might 

 therefore be substituted for the supposition B. But also for the cir- 

 cumference of the circle (1) itself the upper limit of a^ is equal to 

 1. For there corresponds to any arbitrarily chosen number g a 

 prime number p such that at a point on the circumference of that 

 circle having the argument 



(f = 



the argument of 



/=:/ P , i. e. > 



2% 

 that is , differs from jt by less than e for an infinite number of 



P 

 ^•- values. 



