( 484 ) 
the limiting relation between ¢, and «, or in other words the equation 
of the parabola by this substitution in the equation of the straight 
line. So this straight line is a tangent to the parabola, and one 
touching in the point in which also the second limiting value of x, or 
v, coincides with z,. From this follows then this rule. If we draw 
a tangent to the parabola in the area OPQ, then all the points («,,¢,) 
for which one of the limiting values is equal to the value for a of 
the point of contact, lie on this tangent. If we draw a second tangent 
to the parabola, the point of intersection with the first tangent has 
_the property that the values of v of the two points of contact belong 
to it for 2, and a,. If we have drawn one tangent, tangents may 
be drawn from all the points of this line lying on the lefthand side 
of the point of contact, so from all the points for which e, is smaller, 
and ¢, larger than that of the point of contact, to the points for which 
e‚ is larger, and so 2, >,, and the other way about. If we wish 
to indicate in what part of the space OPQ below the parabola the 
points lie for which the values of ¢, and «, are such that the whole 
; ; 1 
closed curve remains restricted either to values of rn or to 
1 | Eten 
values of # << me must begin with finding the point on the para- 
l zetel . 
bola “for ‘which ‘x, = 2, EEN This isthe pomt for which‘ =e. 
and which therefore lies on the line which is drawn from the origin 
in the direction of the axis of the parabola. In this point we must 
trace the tangent to the parabola. From the ¢,-axis this tangent ents 
(n —1)? (n—1)? 
off a portion =-——— and from the ¢,-axis a portion =-—_—__. 
2 2n? 
So it is a line parallel to the straight line PQ of fig. 36, and it 
OP GU a En 
cuts off from the axes parts equal to 7 ane oe This tangent divides 
the space OPQ below the parabola into three parts, viz. the part 
below this tangent, and the two other parts above this tangent and 
further bounded by the parabola and one of the axes. The righthand 
one of these two parts contains the points, for which the closed curve 
1 
remains confined to values of Ht For the lefthand part the 
reverse applies. 
So according to this result either of these cases would be possible 
1 
either that the closed curve remains restricted to values of ae , 
