286.] MOTION IN THREE DIMENSIONS. !$! 



IV. Solid Kinematics. 



I. MOTION OF A POINT IN A TWISTED CURVE. 



285. We have so far considered only those cases of motion 

 where the path of the point is a plane curve. In the most gen- 

 eral case when the path is a so-called twisted or tortuous curve 

 we may refer it to three rectangular axes and resolve the veloc- 

 ity v as well as the acceleration j each into three rectangular 

 components parallel to these axes : 



dx dv r 



-, j,= Jcoa ^- = , 



.-, 



dz . . dv f d^z 



-, ,;=/ = -=, 



i) = V ^ 2 -h 



286. As polar co-ordinates of the point P we take the radius 

 vector OP r, the colatitude xOP = 0, and the longitude 



Q = $, Q being the projection of P on the plane yOz 

 (Fig. 76). 



The velocity z; can be resolved into three rectangular compo- 

 nents : v r along r, v 9 at right angles to r in the plane x OP of 

 the angle 0, and v^ at right angles to this plane. To find their 

 values we take the element PP' = ds of the curve described by 

 the point P as diagonal of an infinitesimal parallelepiped having 

 its edges in those three rectangular directions. The three 



PART I II 



