16 I. KINEMATICS OF A POINT. LAWS OF MOTION. 



Squaring and adding, 31) becomes 



ds * dx 



d8 



_*_> 



2 " h 2 "^ 



\ds z 



Now ,e have 



Differentiating this by s gives 



_ _ 9 idxd*x dy d*y dz 



~ds \ds \ds I ~ (^ds^ds^ '^ ~ds~ds~ ~^~ds 



( dz\ 

 \ds) 



Therefore equation 32) reduces to 



or 



OQN 



33 ) 



If j- = t> = 1, ds = dt and the right hand member of 33) 



becomes -the square of the acceleration. We thus have a kinematical 

 definition of curvature , viz., the acceleration of a point traversing 

 the curve with unit velocity. This agrees with the original expression 



30), a v = ; for if v = 1, a f = 0, the acceleration is entirely normal 



and a v == =x. 



9 



We may in like manner resolve the acceleration into components 

 along the radius vector and at right angles to it. Let us consider 

 the case of motion in a plane, that of XY. We will call the radial 

 component of the acceleration, or the radial acceleration, a ry and we 



d^T 

 shall find that it is not equal to -^-> which is the scalar acceleration 



of the radial velocity. We will denote the component perpendicular 

 to the radius or the transverse acceleration by a^ which is not equal 



to the angular acceleration -=--> nor to the acceleration of the trans- 



verse velocity, ~~- 



Differentiating the formulae for the change of coordinates 



gves 



dx dr . dcp 



dy dr . dm 



