DIFFERENTIAL CALCULUS. 



curvo. Also if two tangents lu> drawn to the curve from a 

 point P lying on the curve (P not being the point of contact of 

 either), the acute angle between these tangents cannot exceed GO . 



9G8. The tangent to the evolute of a parabola at a point 

 where it meets the parabola is also a normal to the evolute. 



x 9 y* 

 9G9. If from a point on the evolute of the ellipse + , t =1, 



the two other normals to the ellipse be drawn ; the straight line 

 joining the point* where they meet the ellipse will bo normal to 

 the ellipse 



970. Prove that, for any curve of the third degree, there 

 exists one point such that the points of contact of the tangents 

 drawn from it to the curve lie on a circle. 



971. A tangent to a given ellipse at P meets the axes in two 

 points, through which are drawn straight lines at right angles to 

 the axes meeting in p : prove that th- normal at p to -the locus 

 of p and the lino joining the centre of tin- ulipsc to the eeir 

 curvature at P make equal angles with the axes. 



972. In a curve of the fourth dr^nv, \\lii.h has four ival 

 asymptote*, no two of whi.'h an- patalJrl, tin- as\ nip toil's w.ll meet 

 the curve again in eight points lying on a conic. Determine this 

 conic in the cate of the curve 



and prov.- that three of the asyn m-h the curve in points 



not at inti 



It tli' piation of a ' :h ?t lh degree be 



and <(c) have two roots //, nnd if also <,(/*,) 0; the cqt. 

 of the corresponding rectilinear ;i 1 bo 



