344 KADIUS OF CURVATURE. 



dr d*r d n r 



. ..XLX be the same for both curves at the 

 ad dti dv 



common point. 



326. Since 4 = 3+ A 



p r r 



it follows from the last Article, that if two curves have con- 

 tact of the first order the value of p will be the same for both 

 curves at the common point. Also, since 



dp 



di) dQ dr d^r 



-/- or -r- involves only r, ,, and - 

 dr dr_ dti d6 z 



~d~e 



it follows that if two curves have contact of the second order 



the value of -?- must also be the same for both curves at 

 dr 



the common point. 



327. We may apply the preceding Article to establish the 

 equation proved in Art. 321 as follows. 



If R be the radius vector of a point in a circle, 

 P the perpendicular on the tangent, 

 c the radius of the circle, 

 b the distance of the centre from the origin, 

 we have, from the properties of a circle, 



Differentiating, c=R -yp . 



If this circle be the circle of curvature at a point in a 

 curve having r for its radius vector and p for the perpen- 

 dicular on the tangent, we have by the last Article, 



P=P> 



