in the higher Geometry. 383 
Scholium. This is a case of a most general enunciation, which 
gives rise to an infinite variety of the most curious porisms. 
prop. ix. porism. Fig. 4. 
A conic hyperbola being given, a point may be found, from 
which if straight lines be drawn to the intersections of the 
given curve with innumerable parabolas (or hyperbolas) of any 
given order whatever, lying between perpendiculars which meet 
in a given point, the lines so drawn shall cut, in a given ratio, 
all the areas of the parabolas (or hyperbolas) contained by the 
peripheries and co-ordinates to points thereof, found by the in- 
numerable intersections of another conic hyperbola, which may 
be found. 
This comprehends, evidently, two propositions ; one for the 
case of parabolas, the other for that of hyperbolas. In the for- 
mer it is thus expressed with a figure. 
Let EM be the given hyperbola ; BA, AC, the perpendiculars 
meeting in a given point A : a point X may be found, such, 
that if XM be drawn to any intersection M of EM with 
any parabola AMN, of any given order whatever, and lying 
between AB and AC, XM shall cut, in a given ratio, the area 
AMNP, contained by AMN and AP, PN, co-ordinates to the 
conic hyperbola FN, which is to be found; thus, the area ARM 
shall be to the area RMNP in a given ratio. 
prop. x. PORISM. 
A conic hyperbola being given, a point may be found, such, 
that if from it there be drawn straight lines, to the innumerable 
intersections of the given curve with all the straight lines drawn 
through a given point in one of the given asymptotes, the 
