DIFFERENTIAL AND INTEGRAL CALCULUS. 113 



or (X, Y) is the same as (a, b). Hence also if be this 

 point (X) I 7 ), P the point (a?, y] at which the normal is 

 drawn, 



the radius of curvature as before. 



DIFFERENTIAL CALCULUS. VI. 



1. What is meant by contact of the m th order between 

 two curves at a point. If two curves have contact of the 

 n h order at a point P, and NQQ' be an ordinate near P, 

 meeting the curves in , Q't the limiting value of the ratio 

 QQ' : PQ n+l when $, Q' move up to P will be finite. If 



two curves have contact of an even order, they cross at 

 the common point ; if otherwise, not. 



2. Explain why a circle cannot in general be made to 

 have with a given curve at a given point a contact of 

 higher order than the second ; and prove that if it have at 

 any point contact of the third order, the radius of curvature 

 will be a maximum or minimum at the point. Prove 

 that a parabola can be found having contact of the third 

 order, and an ellipse or hyperbola having contact of 4he 

 fourth. 



3. Obtain the equations determining the centre and 

 radius of the circle of curvature at any point (a?, y) of 

 a given curve. If 5 be the arc of the curve to (a?, y\ 

 and p the radius of curvature, then will 



d z y d*x 

 1 "2? " ~d? dx d' 2 y dy d*x 



_ ; ..^ = mt ^ m mt m '2. ' 



p dx dy ds ds* ds ds z 

 ds ds 



M) 



