584 



of tlie second. If <f (0 ^vere equal to tlie second tei-ni only, (lie 

 integral (1) could be expanded into a series of factorials for A' (.r) > /i 

 only, and this series would Ue ahsolntehi converging for these values 

 of .1'. Thus ihe lokole funclion may also he represented by such a 

 series for A (.<;)> .". i-^- for A (./) > / and A' (.rj > /', but the con- 

 vergence is, on account of the first term, oidy conditio lut I for 

 X<^ R{x)<:^)' ^ \.. This, again, is exactly the proposition of Niklsen. 



6. If, in the first example of the foregoing paragraph, we account 

 for the reason of the validity of this proposition, we infer that it 

 is a consequence of the fact that the expression 



for a fixed value of / ^ 0, decreases with J//i as the y/-^'' power of 

 a numbei' less than 1, which causes that, in the integral (11), only 

 an interval has to be considered which, in a proper manner, ap- 

 proaches to zero as ii becomes infinite, so that the value of A (.r) for 

 which expansion is possible can be de|)ressed by unity. This suggests 

 the idea that something of the kind might occur (/.v a rale, if Y (0 

 has ?! = 1 for an ordinavy point. The truth of this presum|)fion is 

 proved by the following investigation. 



We again divide the interval (0,J) of t into two partial intervals, 

 with the point (f = r as a common end-point, which is to approach 

 ultimately to zero as n becomes indefinitely large; and we assume, 

 as in the preceding paragraph, for v the value (23). Consider the 

 circle, with centre v and passing through two fixed points C and C' 

 lying on the circumference of the circle of convergence (0,1) of 

 <r [t), symmetrically with regard to the axis of real quantities, and 

 in the interior of an arc D A D' of the latter circle, which does not 

 contain a singular point of <i (/), I) and D' being also conjugate 

 [)oints, whereas A is the point with the affix / = !. Then, from 

 and after some value of n the value of r will be so small that the 

 circle with centre r does not contain any singular point of r/- (^) in its 

 interior and on its circumference; and at all points of the latter 

 between the radii i) D and D, including an arc E B E' of it [B 

 being the point oji that air with argument zero), the modulus of 

 (lit) will remain under a finite quantity A^ independent of ?i and /. 

 As regards points of the supplementary arc E F E' of circle (r), F 

 being the point opposite to B. we may remark that (f it) there has 

 a modulus no greater than 



rf{\-v"), 



