GEOMETRY 



59 



, y'(y'z" - z'y") - x'(z'x" - x'z") 



where 



L = {x' 2 + y' 2 + s' 2 }M (y'z" - z'y"} 2 + (z'x" - x'z") 2 + (x'y" - y'x")*}*. 

 2.677 The direction cosines of the binormal to the curve in the form (b) are: 



/" = y' z " - z 'y" 

 s 



z'x"-x'z" 

 m" = 1 



where 



S = {(y'z" - z'y"} 2 + (z'x" - x'z") 2 + (x'y" - yV') 2 }*. 



2.678 If s, the distance measured along the curve from a fixed point on it is 

 the parameter, /: 



d 2 x d 2 <v d 2 z 



where p is the principal radius of curvature; and 



m fdy d 2 z dz d 2 y\ 

 \ds ds 2 ds ds 2 / 



m "~'fadt~7:'W 



_Jdxd?y_dy**\ 

 P \ds ds 2 ds ds 2 ]' 



2.679 The radius of torsion, or radius of second curvature of a space curve is: 



( x ' 2 4- y' 2 + 2 /2 H 



.// ." " 



"' '" '" 



where 6* is given in 2.677. 

 2.680 When t = s: 



ds 



ds 



