356 CONTACT. CURVATURE. EXAMPLES. 



37. Shew that corresponding to the portion of the curve 

 a*y = a~x* + x- near the origin, the evolute is approxi- 

 mately a curve whose equation is 



38. Shew that the chord of curvature parallel to the axis 



y - 



of x of the curve sec- = e is constant; and that the 

 a 



evolute of this curve for the portion near the origin 

 is approximately a curve whose equation is 



za 



sec ( -*- 1 = e a . 



39. If along a curve and its circle of curvature at any point 



equal arcs (8s) be measured from the point of contact 

 and on the same side of it, shew that the distance be- 

 tween their extremities will be ultimately - -f- ^5- . 



6 as p 



40. Shew that in general a conic section may be found which 



has a contact of the fourth order with a given curve at 

 a proposed point, and shew how to find it when the 

 length of the curve is given in terms of the angle which 

 the normal makes with a fixed line. 



If the curve be an equiangular spiral, and a be the 

 angle between the radius vector and the tangent at any 

 point, shew that the conic section is an ellipse, the 

 major axis of which makes with the normal to the 

 curve an angle o> given by the equation 



tan 2o> + 3 tan a = 0. 



