114 DIFFERENTIAL AND INTEGRAL CALCULUS. 



4. If p be the perpendicular from the origin on the 

 tangent, and </> the angle which the tangent makes with a 

 fixed straight line, then the perpendicular on the normal 



will be , and the radius of curvature = p + -* = T . 

 aq> dtp a0 



Hence, prove that the whole arc of a closed curve without 



,27T 



singular points = I pd<f>. 



J o 



# 2 y 2 



5. The radius of curvature of the ellipse + 77 = 1 at 



a b 



the point (a cos0, b sin#) is (a? sin' 2 #4 b 2 cos' J #)* -^aZ>, and 



the coordinates of the centre of curvature are -- cos :J 0.. 



a 



= sin 3 0. The centre of curvature lies on the curve 



6. If normals to a curve at P, Q meet in 0, the limit- 

 ing position of when Q moves up to P will be the centre 

 of curvature at P. Hence, prove that the radius of cur- 

 vature of a parabola is 2a sec 3 0, 4a being the latus rectum, 

 and 9 the angle which the normal makes with the axis. 



7. In the curve y = \c (e 4- e" ), the radius of curvature 

 at a point where the tangent makes an angle <f> with the 

 axis of x is c sec 2 </>, and if the normal at P meet the axis of 

 x in (7, PC will be equal and opposite to the radius of 

 curvature at P. 



8. The equation of the conic of closest contact at any 

 point of a given curve referred to the tangent and normal 

 at the point as coordinate axes is Ax z -f %Hxy + By* = %y ; 

 the values of A, H, B being 



I L d ? A I JL /^?Y I ^P. 

 p' 3p ds> p 4 9p \ds) ' 3 ds*> 



where p is the radius of curvature at the point and s the 

 arc to that point from some fixed point. 



