200 BOOK OF MATHEMATICAL PROBLEMS. 



98G. The reciprocal polar of the evolutc of a parabola with 

 respect to the focus is a curve whose equation is of the form 



r cos = c sin 2 ; 

 and that of the evolute of aii ellipse is 



c e sin 



rcostf 

 the focus being pole and the axis the initial line in each case. 



987. A rectangular hyperbola, whose axes are parallel to the 

 co-ordinate axes, has contact of the second order with a given 

 curve at a given point (#, y) : prove that the co-ordinates (X t Y) 

 of the centre of the hyperbola are given by the equations 



X-x _ \dx 



. = I -y- jj - : 

 dy d*u 



ilx dx* 



and that the central radius to the point (ce, y) is the tangent at 

 (A. }') to the locus of the centre of the hyperbola. If the given 

 curve be (1) the parabola y* '= ax, ("2) the circle x' + y 2 = a 2 , the 

 locus of the centre of the hyperbola is 



(1) 4 (x + 2a? = 27ay*, (2) x* + y 1 = (2a)\ 



988. A series of rectangular hyperbolas have their axes 

 !lcl to the axes of the ellipse 



and have witli it contact of the second order : prove that the 

 locus of their centres is a curve similar to the evolute of the 

 ellipse, and whose dimensions are to those of the evolute as 



989. A curve is such that any two corresponding points of 

 its evolute and an involute are at a constant distance : prove that 

 the line joining the two points is also constant in direction. 



