348 PHILOSOPHICAL TRANSACTIONS. [ANNO 17Q8. 



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 be- 

 tween 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 2 propositions; 

 one for the case of parabolas, the other for that of hyperbolas. In the former it is 

 thus expressed with a figure. Let em be the given hyperbola ; ba, ac, the per- 

 pendiculars 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. 10. 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 2d conic hyper- 

 bola, 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 inter- 

 sections of the curve with all the parabolas, or hyperbolas, of any given order what- 

 ever, lying between straight lines to be found, the moving line shall cut, in a 

 given ratio, the areas of the curves described, contained by the peripheries and co- 

 ordinates to another cpnic hyperbola, that may be found, at the points where this 

 cuts the curves described. 



Prop. 12. Porism. — A conic hyperbola being given, a straight line may be 

 found, along which if another move in a given angle, and pass through any point 

 whatever of the hyperbola, and if this point of section be joined with another that 

 may be found, the moving line shall cut, in a given ratio, the triangles whose bases 

 and altitudes are the co-ordinates to a conic hyperbola, which may be found, at the 

 points where it meets the lines drawn from the point found. 



Scholium, I proceed to give 1 or 2 examples, wherein areas are cut in a given 

 ratio, not by straight lines, but by curves. 



Prop. 13. Porism. Fig. 14. — A conic hyperbola being given, if through any 

 of its innumerable intersections with all the parabolas of any order, lying between 

 straight lines, of which one is an asymptote, and the other may be found; an 

 hyperbola of any order be described, except the conic, from a given origin in the 

 given asymptote perpendicular to the axis of the parabolas, the hyperbola thus de- 

 scribed shall cut, in a given ratio, an area, of the parabolas, which may be always 

 found. 



