DIFFERENTIAL AND INTEGRAL CALCULUS. 107 



therefore 



BQ =/' 



, 



13- 11 



, RQ f" 



therefore 



or the limiting ratio of RQ : RQ' is/' (x) : <" (x}. 



Hence if two curves touch each other at (a?, y) ; and, 

 therefore, / (a;) = <' (a?), then if also /" (x) = <j>" (x) , the 

 curves will deflect from the tangent at the same rate, or 

 will have the same curvature at the point. This is, there- 

 fore, the same thing as having contact of the second order. 

 This of course is obvious from the expression already found 



for the curvature -r--!l + [ - } \ ; which is the same 



dx* ( \dxj ) \ 



for all curves in which, at a given point (#, y\ -j- and -^ 



have the same values for the different curves. The circle 

 of curvature may then be defined as the limiting position, 

 when Q moves up to P of a circle which touches the curve 

 at P, and passes through a neighbouring point Q of the 

 curve, or since touching at P is the limit of meeting the 

 curve in two points which ultimately coincide, we may 

 also define the circle of curvature as the limit of the circle 

 which meets the curve in three points , P, Q' r when 

 Q, Q' both move up to P. In just the same way any 

 curve which meets a given curve in (w + 1) points of which 

 P is one, will in the limit, when all the points move up 

 to coincidence with P, have contact of the n h order at P, 

 and the order of contact which it will be possible to give to 

 a curve of any defined species will depend on the number 

 of points which suffice to fix such a curve, being one less 

 than that number. Three points completely determine a 

 circle, so that circles can only be made to have contact of 

 the second order. Four points determine a parabola, and 

 we can, therefore, draw a parabola having contact of the 

 third order ; and similarly an ellipse or hyperbola having 

 contact of the fourth ; and an epitrochoid of the fifth order. 



