585 



(p{t) means the natural niajomnt of 7 {t), and i" the (lislanee of the 

 points D and E. 



We further remark that the i-adiiis of the circle (rj is. greater 

 than 1 — V, say J — v -\- v . It is evident that the numbers r' and i'" 

 both approach to zero together with v, but that their ratios to tlie 

 latter number remain Unite and ili[f'ereiit t'roni zero. 



At a point P of the interval (r, J) we have, according to a well- 

 known proposition 



|y^"'(Oj • ^ 



I n! I "^(l—^ + r')"' 



if M is the greater of the numbers K and <p (1 — r"). Instead of 

 this inequality we may write 



1(1 — «)"-lr/ ("){<) 



<'.-^W ''■"-■ 



01 



• «i"ce forO<^<l,^^,<ji- 



\{l—t)»-\(.»)it) 



< 



(T+V)" 



(24) 



Witii regard to rp (1 — r") the following remarks may be made. 

 If, in the equivalence-equation 



Urn a„ — w*' 

 the quantity a' is no less than — 1, we ha\ e, according to the 

 proposition of Cesaró, foi' any fixed ö ^ 



Urn {v"y'+i+^ '7(1—1'") = 0, 



v"=0 



and hence, in virtue of the remark made above on the relation 

 between v" and v, 



li7nv''+^+'q{l—v')=zO 



v=0 



and further 



Urn r^'4-i+» X M=0 



v=0 



since, as a matter of course, the expression [{ X, r'''-^^+^ has, too, 

 zero for its limit. 



Thus we may write for (24), in connection with the assumption 

 (23) and the finite, not disappearing ratio between r and r' 



{■i—ty>-UpW(t)\ 



< 



kn^ 



r{n—l) I ^ (H-An"!-!)" 



