J7 



rdr 2a rdr \' 2 l/a^ — r'n 



I _ lo,f _- = _ - log _ log , 



because 7 is = I /V/)— 1 X ^ '/' — r\ Hence we have for tlie integral : 



Log 



Vk-' 



2 rt r 



— Zo<7 I /or/ 



Ka" — ?•" f/r 



We iiave for tiie last integral with r : a = .v, s : a 



''fi 



dx f x" x' x' 



-^ Jt"- (l-8'n'), 



in wliich s' = 1 for n = '\, and 6 : jr* = 0,(508 for n = (). For 



1 1 1 



r + 4+9 



1 ?r 71* 7}^ 



6 "'' '^"^ T + 4 + 9 + 



71' 71' 



: nM 1 -] \ f . . 



i8 = Y ^' for 7z = 1. and = n' for 7i = 0. (For n = 0,6 5' = 0,674). 



Hence we get (inallj^ : 



1 



(cp > (f,) a 



2h(1— n^) 



(^/)oo« 



— :rt' (l—e'?i') + loei ^ log— ^_,=^= 

 -^2 «^ l/y('V-l_ 



.(17) 



When we compare this with (14''), where we found for values 

 of (f in the neighbourhood of (f^ (but <^ (f^, wliile <f remains 

 >y„ in (16)): 



{<P < <fo) a = 



1 



2w(l-jr) 



^(^..)x« 



"1 112 



J jr^ (1 -f ?i) -f log^ - ^ log~^ log 



^^l-Ff/_ 



we observe with regard to the member that is independent of 2\ a 

 discontinuity appearing at rp =: cp^. [We have added, for a comparison, to 



the first (finite) term the term log'-, which was cancelled in ^18 in 



71 



form. (136) by the side of the infinitely large logarithmic lermj. 



1 



For 71 =: 1 we find (with the factor from the factor before the 



1 - n' 



... 1 1-^*' 1 



sign of integration) in the first case — jt^* = — jt\ in (he second case 



12 1 — 7i' 12 



1 1-w , 1 1 - 1 



— -TiV -{- log^ - =z -- ji\ And for 71 ^=z we find — .-rV resD. 



41 — n* ??8 12 



111^ 



— - jr* -f log^ - = -- .^"^ -f- <^'- This difference can be partly accounted 



for by the sudden disappearance at (p =z (f^ of the terms which refer 



2 



Proceedings Royal Acad. Amsterdam, Vol. XXI. 



