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&amp;lt; in 

 the direction of the tangent at P and v sin Ac in the direction of 

 the normal at P. 



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

 the limit of . ^ when A is indefinitely diminished. In 



passing to the limit cos A&amp;lt; 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 -j- or v. Since we have 

 at 



dv _dv ds _ dv 

 dt == ds di~ V ds &amp;gt; 



we may write v -7- for the component acceleration parallel to the 

 ds 



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



Again the acceleration in the direction of the normal at P 

 is the limit of ~AT~ &amp;gt; an&amp;lt; ^ this is the same as the limit of 



v sin A&amp;lt;^&amp;gt; A^&amp;gt; As 



v A$ As A 



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 



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 



dx dy dz , . /dx\ 2 fd&amp;gt;/\ 2 /dz\* 



increases, are -=- , , -=- , satisfying the relation U- + U- + U =1 ; 





