( 493 ) 
If we call the value of a required to change the inequality into 
equality for given value of y, « — then the relation: 
27 1— 1—y)’ 
ee ee at y) 
4 (l+y)° Ay? 
holds for this quantity. 
For the preceding problem, viz. the determination of the relation 
3 dw 
d 
between « and y causing —=0 to disappear on the curve T= 0, 
& v~ 
dy—l 
Ay? 
1 —a= 
held. 
For a@'—a we find then: 
: lty 27 1—y 
a'—a= pat ee == 
4y? 4 (1+y)’ 
or 
(ten (veer (EN orice 
4y* (149) uP (ty 
From this it appears, what had been clear beforehand, that a’ is 
eo — 
1 
always greater than «,‚ except for y= oe when they are both equal 
to 0, and so for the points of the tangent. A case, however, which 
we can only think as a limiting case, because it would require 
n=o. The adjoined figure 38 gives the relation between «a and x 
for the two problems graphically. For the origin «= 1, and for the 
points of the tangent ¢=0O. For the first problem vz for the 
origin, and for the second y= 0,36 — whereas for a=0O the two 
values of y are =>. For the second problem the line y= f(«) 
always lies above that of the first problem. Hence for equal value 
of y the point P’ lies at higher value of a than the point P. 
(To be continued). 
33 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
