941 



For this purpose, we need only choose p so great that /38^ jr. The 

 congruence 



2^ = 1 (mod. p) 



can be satisfied according to a theorem of Fermat for all /(.-values 

 being a mulliple of />— 1 ; that is, for all ihose values we have 



2^ = (2/ -f l)p + 1 

 or 



2% jr 



P P 



so that in connection with the above choice of the number p the 

 required condition is satisfied. Thus there are always points on tlie 

 circumference of the circle (1) for whicli the quantity 



differs from the value 2 by less than an arbitrarily small amount 

 so that it is for instaiice greater than 1. Since in a,„ {-i;) ^ x"-'^ the 

 number n assumes all integral powers of 2 as a value, a„i (.v) is in 

 the just-mentioned points for an infinite number of 7/2,-values greater 

 than 'J, so that ^a; = 1 in all those points. 



We have thus constructed an example in which the conditions 

 A and B are satisfied, but the uniformity-supposition of N° 4 is not 

 satisfied. As, now, regards the two former suppositions, we should 

 like to have an example in which either one of them or both were 

 not realized. But we have not succeeded as yet in constructing 

 any of the kind, nor, on the contrary, in proving that this would 

 be impossible, in which last case A and B would hold univer- 

 sally. If there be a point on the circumference of the circle («) 



1 



such that at that j)oint the quantity | a,n |'" be for an infinite number 

 of ?/?--values, m^, m^, . . . in.n, . ■ . equal to the maximuui 



1 



aZ (<^) 



of the same quantity on the circumference of [a), and if at the same 

 time the upper limit of the partial sequence 

 1 1 1 



aZ\ («) , aZ', («) , . . , .4"!;; {«) , . . . 



be equal to that of the complete one, then at the point mentioned 

 ax = A («), and thus a («) = A (n) so that both A and B are satisfied. 

 In constructing a pathological example as mentioned we should therefore 

 take care that there is no such [)oiut on the ciicumfereuce of 

 (a). But this is by no means sufficient. For it may be possible that, 



66 

 Proceedings Royal Acad. Amsterdam. Vol. XX. 



