400 MISCELLANEOUS EXAMPLES. 



26. In any curve the equation ~ + l = Q holds at a point 



of inflexion, 6 and < being the angles which the prime 

 radius and tangent make respectively with the radius 

 vector. 



dv 



27. Is -JQ necessarily of the form - at a multiple point ? 



Ccv \J 



28. Find the singular points in the curves 



and 2/ 2 - 2xy + 2x* -a? = Q. 



29. Find the nature of the curve 



y + 1 = 2x - a? (2 - a;)* 

 at the point x = 2. 



30. Determine the point of inflexion in the curve 



31. From the pole of the curve r = Aa e perpendiculars are 



drawn upon the tangent ; through the points of inter- 

 section of the perpendiculars with the tangents, straight 

 lines are drawn parallel to the radii vectores : shew 

 that the equation to the locus of the ultimate intersec- 

 tions of all such straight lines is r = A cos aa^" a , where 

 cot a = log a. 



32. If radii vectores of an equiangular spiral be diameters of 



a series of circles, the locus of the ultimate intersections 

 of the circles will be a similar spiral. 



