VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 347 



be found, lying between the lines drawn at right angles to each other, the lines so 

 drawn from the point found, shall be normals to the parabolas at their intersections 

 with the ellipse. 



Prop. 4. Porism. — A conic hyperbola being given, if through any point of it a 

 straight line be drawn parallel to the transverse axis, and cutting the opposite 

 hyperbolas, 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 hyperbolas of 

 orders to be found, lying between straight lines which may be found, the straight 

 lines so drawn shall be normals to the hyperbolas at the points of section. 



Scholium. The last 2 propositions give an instance of the many curious and ele- 

 gant analogies between the hyperbola and ellipse ; failing however when, by 

 equating the axes we change the ellipse into a circle. 



Prop. 5. Local theorem. Fig. 11. — If from a given point a, a straight line de 

 move parallel to itself, and another cs, from a given point c, move along with it 

 round c ; and a point i move along ab, from h, the middle point of ab, with a 

 velocity equal to half the velocity of de ; then, if ap be always taken a 3d propor- 

 tional to as and bc, and through p, with asymptotes d'e' and ab. a conic hyper- 

 bola be described ; also with focus i and axis ab, a conic parabola be described ; 

 then the radius vector from c to m, the intersection of the two curves, moving 

 round c, shall describe a given ellipse. 



Prop. 6. Theorem. — A common logarithmic being given, and a point without 

 it, a parabola, hyperbola, and ellipse, may be described, which shall intersect the 

 logarithmic and each other in the same points ; the parabola shall cut the logarith- 

 mic orthogonally ; and if straight lines be drawn from the given point to the 

 common intersections of the 4 curves, these lines shall be normals to the lo- 

 garithmic. 



Prop. 7. Porism. — Two points in a circle being given, but not in one diameter, 

 another circle may be described, such, that if from any point of it to the given 

 points straight lines be drawn, and a line touching the given circle, the tangent 

 ehall be a mean proportional between the lines so inflected. Or, more generally, 

 the square of the tangent shall have a given ratio to the rectangle under the in- 

 flected lines. 



Prop. 8. Porism. Fig. 12. — Two straight lines ab, ap, not parallel, being 

 given in position, a conic parabola mn may be found, such, that if, from any 

 point of it m, a perpendicular mp be drawn to the one of the given lines nearest 

 the curve, and this perpendicular be produced till it meets the other line in b ; and 

 if from b a line be drawn to a given point c ; then mp shall have to pb together 

 with cb, a given ratio. 



Scholium. This is a case of a most 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 



Y Y 2 



