189 



Thi' v-rurvcs are also plane runes. 



1 '/^\' /^'V /^'V 



Torsion = = A(i~~li"( — l"f ( — I where /, //, r are direction 

 T \Vds/ Vds/ Vds/ 



cosines of bi-normal to curve (here X2, Y2, Z2) 



dX2 dX2 'iXo dY2 dZz 1 



- - =r — = — = o and similarly = o = — . •. = o. 



dsv du '^u ds ds T 



The u-curves have constant radius of curvature. 

 The equation of radius of curvature is, 



/j2 Vds2/ Vds''^/ \ds-v 



dx dx 



— = = X2 



ds ds» 



d2x d 



— = — X2 

 ds2 dsu 



'5X2 dv 



'5v dsu 



1 <5X2 _ 



= , since dsn = y G dv 



i/G <W 

 d^x 1 / N d^/G 



1 / N d^/G \ 

 -(— X x) 



. G \, G du / 



dV 1 / N d,^ \ 



So, --= -^(— lY Y,) 



ds2 |/G\i/G du / 



d^z 1 / N dv/G \ 

 - = — (— Z -Z.) 



ds2 , gV,/G du / 



I 1 rN= N /dv G\' 1 



- = — I _ IX^ — 2 — - IX , X ^ I 1 IXJ I 



)2gLg iG Vdu/ J 



1x2 = 1 = vxi + ^x, X = o 



,' G L<^ V du / J 



l> 



= function of w^ only 

 const for u-curves. 



