( 735 ) 



1 db da 1 d*a . 



significance, requires — — — > — — - . r rom the foregoing remarks it 

 b* dx dx 26 dx' 



is sufficiently clear that -- must be positive to render the locus I— dv 



dx J dx 



d*a d*\p 



= possible, and that — must be positive to render = possi- 



dx* dx 2 



ble. The roots of the given quadratic equation, however, are then 



1 db da 1 d*a 



imaginary, the square of — - — : — -7-, being necessarily smaller 



b* dx dx 2 b dx 



1 db da 



than the square of — , and the square of this being smaller 



b dx dx 



than the product of — ( — J and the factor of (MRT) . 

 b \dxj 



But let us return to the description of the course of the remaining 



^-lines. There is, of course, a highest ^-line, which only touches the 



d*tp 



locus = 0, directed horizontally in that point of contact, and 



dx* 



d*\p 

 for which also — - = in that point. There is also a (7-line which 

 dx z 



touches the locus = in its downmost point, and which as a 



dx* 



rule will be another than f hat which touches it in its highest point. 

 The g-lines of' higher degree than the higher of these two have again 

 the simple course which we have traced in fig. 2 (p. 635) for thai 



r/-line which intersects the locus ( — ) = 0. Only through their eon- 



\dxj 



siderable widening all of them will more or less evince the influence 

 of the existence of the above described complication. The g-lines of 

 lower degree than the loop-^-line have split up into two parts, one 

 part lying on the left which shows the normal course of a ^-line 



which cuts ( — J = Q j and a detached closed part which remains 



enclosed within the loop. Such a closed part runs round the second 



fd P \ d*\\> 



point of intersection which — ) = and = have in common, 



\dxj v dx* 



( /> 



passes in its lowest and highest point through = 0, and 



dx* 



fdp\ 

 through I — = in the point lying most to the right and most to 

 \dxj v 



the left. With continued decrease of the degree of q this detached 



