2750 



PLANE MOTION. 



153 



This angle is therefore called the angle of relative contingence; 

 the quotient dO/da- = (da' da)/ da, where da=CC v is called 

 the relative curvature, and the reciprocal value, da/d6, is the 

 radius of relative curvature. 



Nowo> = d$/^, u = da/dt; hence 



a) _dd __da' da 

 u da da 



or as da I da = l/p, da 1 /da= i/p f , 



co dO I I 



u da- 



(3) 



i.e. the ratio of the angular velocity to the velocity of rolling is 

 equal to the relative curvature of the centrodes. 



Fig. 70. 



When p<p', that is when da>da', the relative curvature is 

 negative. When the centrodes lie on opposite sides of the 

 common tangent we should find in absolute value dd = da' + da. 

 But taking into account the sense of the angles da, da' we still 

 have d0 = da' da. The formula (3) holds, therefore, generally 

 if the radius of curvature p of c be taken as positive or nega- 

 tive according as it lies on the same side of the common 

 tangent with the radius of curvature p f of c', or on the opposite 

 side. 



