354 CONTACT. CURVATURE. EXAMPLES. 



21. KyW*-q, thenp=^-^. 



*/ 



22. Shew that the circle iC ~x 



curve A# + V = V have contact of the i/aVd5 order at 



the point x = y = - . 



/? 9 



23. Ifr = osec 2 ^, find p. esft. p = 2asec 3 -. 



24. Find the two parabolas which, having their axes parallel 



to the co-ordinate axes respectively, have a contact of 

 the second order with the circle ce 2 + y s = bd\ at the 

 point ac = a, y = 2a. 



Results. 



( Sa\' 2a (la \ ( a\* 16a/lla N 

 . (y--)^-(^- X ^ (x - -) = - ( - y) . 



v 1 - - - 

 25. In the curve - = - (e e + e ), shew that the co-ordinates 



C 2i 



of the centre of curvature are 



and find the equation to the evolute. 



26. Find the equation to the evolute of the ellipse, and the 



whole length of the evolute. 



Results. (ax}* 



27. If T =f(p) be the polar equation to a curve, shew that 



the equation to the locus of the foot of the perpendicular 



r* 

 drawn from the pole on the tangent is p = . Find 



JV Jy> 



the locus when 2 = - -- , and shew that it is a circle. 



r 



