9, 10] TANGENTIAL AND NORMAL ACCELERATION. 15 



,. AT dr 



lim = = x. 



j,=v As ds 



Since the angle between the tangents is equal to that between the 

 normals 



ds = Qdr 



29) ;==- 



If as before we draw lines AB, AC (Fig. 6) representing the 



T> (~l 



velocities at P and Q, the acceleration, lim , is in the plane of 



AS and AC, that is the vector acceleration is in the osculating 

 plane. As we have already found the component parallel to the 

 tangent, there remains only the component parallel to the principal 

 normal. Since BC is proportional to the acceleration, DC is pro- 

 portional to the tangential acceleration a t , BD to the normal 

 acceleration a vy and since the angle at A is dr and the side AB is v 9 



BD = vdr 



vdr 



Also since ds = gdr 



OA \ v ds v' 2 



30 ) a= -==. 



gdt <? 



This normal acceleration is always directed toward the center of 

 curvature, and is otherwise called the centripetal acceleration. 

 Inserting the above values in the equation 



a 2 = al + al, 

 we may obtain an analytical expression for the radius of curvature. 



Let us change the independent variable from t to s. We have 



dx _ dx ds 

 ~di ~ds^li' 



and differentiating again by t, 



d*x_dxd*s . d* 

 ~di*~~ lte~di* + ~ds 

 and similarly 



d*y ___ dy d 2 s d 2 y ( ds\ 2 

 ~dt^ ~ ds ~di*~ + ~Js* dt) ' 



