140 



The u-curves are also plane curves, and therefore circles. 

 We may write equation of torsion inlthe form, 



1 i x' y' 7/ i 



— = — /'- I x'^ y" 7." I 

 T I -Kf'' y'" 1.'" I 



(where primes denote derivatives with respect to s). 



From last paragraph we have^for u-curves, 



d^x 1 f5X, 



ds' 



X'" = 



ds 



^/ G '^v 



1 '5X^ 



d / 1 '5X.A 



.s„ V^/G 'V / 



(I <^XA 

 ./G '5v / 



'iv V/G '5v 



1 'i^x. 



dv 



ds„ 



G 'W2 

 1 FF - 



G LG 



tl'" = ^(u) X, 

 Soy''' = 0(u) Y, 



Z'"^=r 9(U) Z, 

 1 1 



— — — r''^(u) -— 

 T i/G 



[^(^)1 



Xjfrom (6), (4) and (8) 



X2 Y^ Z2 



r(X , r5Y , rfZ , 



fSv '^v f'v 



X, Y, Z, 



= 



Since the u-curves are plane and have constant radi of curvature they 

 are circles. 



Finally, the plane of each v-curve is normal to every u-circle, and 

 therefore passes through its center. The intersection of any two v-planes 

 determines the line of centers of the u-circles. Thus all the required sur- 

 faces are surfaces of revolution. Taking the line of centers of u-circles as 

 z-axis and the plane of any u-circle as xy-plane, the equation of our sur- 

 faces are 



X :;= u . cos v 



y = u . sin v 



"z ^ f(v) 



