( 525 ) 



e z=: l,bü 

 2,72 

 4,48 

 12,2 

 20,1 



' = 33,1 

 54,6 

 148 

 1810 

 22000 



4,14^ „ 



4,08 „ 



4,01 „ 



4,00 „ 



Reallv the maxiiiiuni lies just })ast ry^r=67\. (We saw already 

 above, that for q, = 47\ also q^ = 4:1\). 

 b. The curve II, viz. 



2 {q, — 4:T^ 

 4r, + -^ -TT^. 



(8^^) 



1 + e-^ 



This curve separates the curves T = f{,x) with concave end {left 

 of this curve, because q^ is then smaller than the second member) 

 from that with a convex end {;right of ihe curve, where q^ is largerj. 



For ^1 = also q^ ■= 0, as ^j = 0; {initial direction again 

 q^ =r q^ (45°) ) ; for q^:=z4T^ also ^, = 4T,, and for 7^ = 00, ^^ will 



approach to 2 r/j — 47",, because ö~ ' approaches to 0. The limiting 



direction of the curve II is therefore given by ^^ =2^'^, or ^1 = V2 9'2- 

 (26=,5). 



It will necessarily cut I. When T^ = Va^n this point of inter- 

 section aSj lies somewhat on the left of the maximum j\I\. It is 

 found by combining 



?i = 4T, H -^ and q, = 2T, H — — . 



1 + ^ I + "3 



By approximation we find q^ = 5,907\, ^j = 4,19Tj . 



The further calculation leads to the following summary. 



—9 



ITAe ^ = 0,61 



3 „ 



4 „ 



5 „ 



6 „ 



0,22 

 0,13= 

 0,08 

 0,05 



q,=8T, 

 10 „ 

 15 „ 

 20 



e~ ' — 0,02 

 0,01 

 0,00 

 0,00 



^,=:13,8r, 



17,9 „ 

 28,0 „ 

 38,0 „ 



For q, — 2T, {— 47\) also q, = 22\ (see above) 

 c. The curve III, i.e. 



- 47\ + 



2 (93 + 47\) 

 1 + e-'' 



(8111) 



