289.] MOTION IN THREE DIMENSIONS. 163 



at right angles to this plane, and collecting the terms, we 

 obtain : 



. fidr d$> 

 dt* ' dt dt 1 '"dt dt 



288. It is to be noticed that the resolution of the accelera- 

 tion/ into a tangential component/ and a normal component/,, 



_^ _ ^ 

 dt' p' 



given in Art.. 159, holds for twisted curves as well as for plane 

 curves, provided the normal be understood to mean the prin- 

 cipal normal of the curve, and p the radius of absolute curvature 

 at P. For it follows from the definition given in Art. 155 that 

 the acceleration lies in the plane of the tangent and principal 

 normal at P, so that the component along the binormal is zero. 



289. This can also be seen from the expressions for the com* 

 ponents of/ in Cartesian co-ordinates, j x = d*x/dt*, j y = d*y/dt*, 



j g =d*z/dt*. For since ^=^ ^ etc., we have 



dt ds dt 



dx . fds\* d*x 



= 

 Js dt* dt* ds \dt 



= ^ , 



Jy dt* dt* ds \dt ds*' 



. = d*z^d*s<te (ds\*d*z 

 Jz dt* dt* ds \dt)ds* 



Now, dx/ds, dy/ds^ dz/ds are the direction cosines of the 

 tangent of the curve, while pd*x/ds* y pd*y/ds*, pd*z/ds* are. the: 



