35, 36] CURVED PATH. 37 



the direction cosines of the principal normal directed towards the centre of 

 curvature are p -r% , p -j\ 2 , p -3-2 satisfying the relation 



p 



where p is the radius of circular curvature ; and the direction cosines of the 



/d 2 y dz d 2 z dy\ fd 2 z dx d*x dz\ (d^x dy d*y dx\ 

 binormal are p I -rfe -^ --- j-5 ) ? P 1 -TO ~r --- r^r ~r I ) P ( ~7l&amp;gt; ? ri TT ) 

 r \flfo 2 cfo cfo 2 cfo/ H \o?5 2 c?5 ok 2 / p \rfs 2 rf* ds 2 ds / 



. . . dx d^x dy d 2 y dz d 2 z _ 

 We recall also the relation -=- -=- 9 - + -f ~A+-r T^ = O - 

 ds ds 2 ds ds 2 ds ds 2 



In the expressions #, J/, z for the component accelerations parallel to the 

 axes we change the independent variable from t to s. 



We have, writing v for the speed, so that v stands for s, 

 ..d 2 x d fdx\ds d (ds dx\ d / dx 



dv dx d 2 x 



so that x=v -j- -T- + V* ~Y^ , 



ds ds ds 2 



dv dy d 2 y 



so y= v ^T -^r + v -339 



ds ds ds 2 



dv dz d 2 z 



and ***-j-T&quot;H^j-5 



ds ds ds 2 



If we multiply these component accelerations in order by the direction 

 cosines of the tangent and add, we obtain the component acceleration parallel 

 to the tangent to the curve in the sense in which s increases ; we thus find 

 for this component the expression 



dv ~dx\* d* fdz\r . dx d 2 x d d 2 dz d 2 z\ dv 



Again, if we multiply by the direction cosines of the principal normal and 

 add, we obtain the component acceleration parallel to the principal normal 

 directed towards the centre of curvature ; we thus find for this component 

 the expression 



dz d 2 z~ Vd 2 x\ 2 cP\* d 2 z 



p- 



Finally, if we multiply by the direction cosines of the binormal and add, 

 we find no component acceleration parallel to the binormal. 



Thus the acceleration of a point describing a tortuous curve is in the 

 osculating plane of the curve, and its resolved parts parallel to the tangent 



and principal normal are v and - , exactly as in the case of a point describing 



CvS p 



a plane curve. As in that case, the expression for the former component 

 may be replaced by v, or by s. 



