139 



Thi' v-rurres are a/.s-o plane cunes. 



1 l,d/^2 d/.x2 /dr. 2 



Torsion ==: *( — 1 + ( — |~t ( — I where /, ,", i' are direction 

 T Wds/ Vds/ Vds/ 



oosiues of bi-normal to curve (liere X2, Y2, Z2) 



dX2 dX2 'iX2 dY2 dZ2 1 



— = — = — ^ o and similarly = o = — . •. =; o. 



dsv du '^u ds ds T 



The u-curves have constant radiux of curvature. 

 The equation of radius of curvature is, 



1 ^d2xv2 ,d2y>.2 d2Zx2 



,,2 \ds2/ Vds2/ \ds2/ 



dx dx 



ds dsu 



d2x d 



— = — X2 

 ds2 ds„ 



'5X2 dv 



'iv dsii 



1 f5X2 _ 



= — , since dSu = y G dv 



|/G <^v 

 d2x 



1 / N dv/G \ 

 — (— X x) 



i/G Vy/G du / 



1 / N d, "G \ 



-( — Y yA 



^G\, G du / 



1 / N dj/G \ 



— (— z z.) 



rGVy/G du / 



1 1 rN= N /dv/G\= 1 



- = - - -X - 2 — - -X, X -r- I 1 ^X^ 



.2 G Lg I G V du / J 



:i;X2 = 1 = i;xj + -X 



d82 |/u- N] 



d2y 1 / N d,'G 



So, -1 = ^ 



ds2 i/G^i G du 



d2z 1 / N _ dj/G 



ds2 



I 

 liX^ = 1 = i;XJ + -X, X = o 

 1 



i>^ G 

 ' = function of U'^ only 

 const for u-curves. 



