458 Koikes respecting New Books, 



5. A normal at any point of an equilateral hyperbola is equal 

 to the distance of that point from the centre. Required the de- 

 monstration. 



6. C is the centre of an ellipse and F either focus^ PH a tan- 

 gent at P : draw the diameter PCp and pF meeting PH in H. 

 Prove that pH = the transverse diameter, 



7. P is any point in the diameter of a circle and PB a per- 

 pendicular to the diameter: draw any chord PA, and a tangent 

 AB meeting PB in B, and drawBD and CE (C the centre) per- 

 pendicular to PA; then PE = DA. Required a demonstration. 



8. Within a given triangle suppose another triangle to be in- 

 scribed, by joining the middle points of its sides ; and again, 

 within this triangle suppose another triangle to be inscribed by 

 joining the middle points of its sides, and so on ad vrfinitum : 

 What will be the limit of the aggregate of the sum of the squares 

 of all the sides of all the triangles so formed ? 



9. Let any right line be drawn through the focus of a given 

 conic section, terminating in the curve; then a fourth proportional 

 to the whole line and the two segments thereof, made by the fo- 

 cus, will always be of the same constant length. Required a de- 

 monstration. 



10. A triangle being given, it is required to describe three cir- 

 cles, so that each circle shall touch the other two and a side of 

 the triangle at the point of bisection. 



11. Required the curve that has at each point the radius of 

 curvature a fourth proportional to the abscissa, the ordinate, 

 and a given straight line, 



12. To determine the nature of the curve such that the per- 

 pendicular from a given point upon the tangent shall be a mean 

 proportional between a given line and the segment of the axis in- 

 tercepted between the tangent and this same given point. 



13. To determine the equation to the curve whose tangent is 

 a mean proportional between the segment of the axis intercepted 

 between it and a given point, and that same segment augmented 

 by a given line. 



14. A body, urged by a force perpendicular to the horizon, de- 

 scribes the quadrant of a circle. Required the law of force which 

 will make it recede uniformly from the horizontal radius, and 

 the time elapsed and the velocity acquired at any point of the de- 

 soent. 



15. The characteristic property of the circle is, that all the 

 chords which pass through a certain determinate point in its plane 

 are equal : but there exists an indefinite number of curves which 

 possess this property. It is required to find the most general 

 equation to curves of this nature. 



16. A given rod or beam has one end suspended by a cord of 



a given 



