( 52G ) 



For values of q^ larger than the second member the beginning of 

 2'=zf{x') is concave; these curves lie therefore above the curve. In 

 the same way all the lines 7'=: ƒ(,!?') with convex beginning lie 

 belo\v this cur\e. 



Again '7i=0, Avhen ^,=0 {initial direction 'Zi—^, (45°)). When 9, 

 approaches to oc, q^ approaches to 2 ^^ -f 42\, 80 the limiting direc- 

 tion becomes q^ = 2 q^ (63°,5). This curve lies entirel}' outside the 

 two first, more to the left. 



Some points of tiie curve III follow. 



-% 



= n\e ' = 0.61 



2 „ 

 4 „ 

 6 „ 



0,37 



0,13' 



0,05 



Sl\\e ' = (),{)2'q, =r 19,6 1\ 

 10 „I 0,01 i 23,8 „ 



(/. T/te curve IV, i. e. 



'h = - 4T, + 



15 „ 



20 „I 



2 (7: 4- 4rj 



0,00 

 0,00 



34,0 „ 

 44,0 „ 



(8^n 



1 + /' 



If q.j is smaller than the second member, the end of T = f{x') 

 will be concave; these lines lie accordingly left of the curve; on 

 the right the lines 7'i=/'(,r') with convex end are found. 



For 7i = again q., = (initial direction q^ = 7, (45^^)). If ^^^ ■=■ co, 



72 evidently approaches nsi/niptoticaUy to ^j = — 42',, just as the 

 curve I approached asymptotically to ^i ^= 47'^, when q., = co. Tlie 

 cui-ve IV lies therefore only for a small part within the region of 

 the positi\'e 7,, and will therefore necessarily cut the ^^-axis some- 

 where in S.i, and yield a ma.rimion \'aiue J/, for 7.^ before that time. 

 This curve too lies therefore entirely outside tiie preceding curves, 

 and again more to the left. 



The 7j-axis is cut, when (jf', = \\ T^) 



or when 



This is satisfied by 



1 f ^ 



^'/sr, 



= T,, 



— 2 



27', 



= 1. 



-|p = l,25', or 7, = 2£17\. 



