36 VELOCITY AND ACCELERATION. [CHAP. III. 



between the tangent at P and the tangent at Q. Also let A* 

 be the time taken by the point to move from P to Q, and let 

 As be the length of the arc PQ. 



The velocity at Q can be resolved into components v' cos A< in 

 the direction of the tangent at P and v' sin A< in the direction of 

 the normal at P. 



Hence the acceleration in the direction of the tangent at P is 

 the limit of - 008 A< ^^ when A* is indefinitely diminished. In 



passing to the limit cos A< differs from unity by an infinitesimal 

 of the second order, and v differs from v by an infinitesimal of the 

 first order, viz. by the limit of Av the increment of the velocity. 



Thus the above limit is -^ or v. Since we have 

 at 



dv dv ds dv 



we may write v -^ for the component acceleration parallel to the 



CiS 



tangent, and we may also write s for it, since v is s. 



Again the acceleration in the direction of the normal at P 

 the limit of - ~- , and this is the same as the limit of 



s 



v sin A0 A</> As 



v V A< As ~Kt ' 



and the limits of these factors in order are 1, v, 1, - , v, where p is 



the radius of curvature of the curve at P. Thus the acceleration 



v 2 



in the direction of the normal is . 



P 



*36. Acceleration of a point describing a tortuous curve. We add 



here an investigation of the acceleration of a point describing a tortuous 

 curve which occupies a fixed position with respect to the axes. 



We recall the facts that if #, y, z are the rectangular coordinates of a point 

 of a curve and s the arc measured from some particular point of the curve to 

 the point (#, y, z\ the direction cosines of the tangent, in the sense in which s 



increases, are ^, |, *, satisfying the relation g?)'+ (| 



