IN THE HIGHER GEOMETRY. 11 



Scholium. This is a case of a more general enunciation, 

 which gives rise to an infinite variety of the most curious 

 porisms. 



PROP. 9. Porism. Fig. 13. A conic hyperbola being given, 

 a point may be found, from which if straight lines be drawn 

 to the intersections of the given curve 



lg.13. 





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with innumerable parabolas, or hyper- B 

 bolas, 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 of * 

 it, found by the innumerable 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 former it is thus expressed with 

 a figure. Let E M be the given hyperbola ; B A, A c, the per- 

 pendiculars meeting in a given point A : a point x may be 

 found, such, that if x M be drawn to any intersection M of E M 

 with any parabola A M N, of any given order whatever, and 

 lying between A B and A c, x M shall cut, in a given ratio, the 

 area A M N p, contained by A M N and A p, P N, co-ordinates to 

 the conic hyperbola F N, which is to be found ; thus, the 

 area ARM shall be to the area R M N P in a given ratio. 



PROP. 1 0. 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 first-mentioned lines shall 

 cut, in a given ratio, the areas of all the triangles whose 

 bases and altitudes are the co-ordinates to a second conic 

 hyperbola, which may be found, at the points where it cuts 

 the lines drawn from the given point. 



PROP. 11. Porism. A conic hyperbola being given, a 

 straight line may be found, such, that if another move along 

 it in a given angle, and pass through the intersections of the 



