MISCELLANEOUS EXAMPLES. 399 



19. The equation to a parabola referred to any two tangents 



being [- ) + (f V = 1, shew that the radius of cur- 

 W \bj 



vature is = : - fax 2 cos a tJlabxy] + by}?, where a 

 l 



is the inclination of the tangents ; and thence find the 

 co-ordinates of the vertex assuming that the curvature 

 is a maximum at that point. 



20. If a curve pass through the origin and touch the axis 



of y, the diameter of the circle of curvature is equal 



V 2 



to the limit of *- : if it touch the axis of x the diameter 

 x x s 



is equal to the limit of . 



y '. 



21. If a curve pass through the origin at an inclination a to 



the axis of x, shew that the diameter of curvature at 



a? 4 y 2 



the origin is the limit of r-^ " - . Hence, shew 

 x sin &y cos a 



that the radius of curvature at the origin of the curve 

 if + 2ay 2ax = is 2 \/2a. 



22. If (j> be the angle between the tangent and the radius 



vector of a polar curve, shew that the radius of cur- 



. r cosec <> 

 vature is 7 . . 



23. The equations to an epicycloid being 



x = a(2 cos 6 cos 20), 

 y = a (2 sin 6 sin 20), 



shew that p = - sin - , and that the evolute is an epi- 

 o 2 



cycloid in which the radius of each circle is - . 



24. In the curve y =x 4 4a; 3 18x*, find the nature of the 



curve at the points x = 3, 1, and f (1 + \/5). 



25. Shew that the curve y e~ x * has points of inflexion when 



