in the higher Geometry. 389 
it, there is a deficiency in the hypothesis : neither is it a 
porism ; for the theorem, from which the deficiency distin- 
guishes it, is pot local. 
prop. xv. porism. Fig. 6 . 
A conic hyperbola being given, two points may be found, 
from which if straight lines be inflected, to the innumerable 
intersections of the given curve with parabolas or hyperbolas, 
of any given order whatever, described between given straight 
lines, and if co-ordinates be drawn to the intersections of these 
curves with another conic hyperbola, which may be found, the 
lines inflected shall always cut off areas that have to one 
another a given ratio, from the areas contained by the co- 
ordinates. 
Let X and Y be the points found ; HD the given hyperbola, 
FE the one to be found ; ADC one of the curves lying between 
AB and AG, intersecting HD and FE : join XD, YD; then 
the area AYD : XDCB in a given ratio. 
prop. xvi. porism. Fig. 7. 
If, between two straight lines making a right angle, an infi- 
nite number of parabolas, of any order whatever, be described, 
a conic parabola may be drawn, such, that if tangents be drawn 
to it at its intersections with the given curves, these tangents 
shall always cut, in a given ratio, the areas contained by the 
given curves, the curve found, and the axis of the given curves. 
Let AMN be one of the given parabolas ; DMO the parabola 
found, and TM its tangent at M ATM shall have to TMR 
a given ratio. 
