140 



Thf u-cunes are also plane cwves, and therefore n'rrlei'. 

 We may write equation of torsion injthe form, 



T 



(where primes denote derivatives with respect to s). 

 From last paragraph we have^for u-curves, 



1 <iX,. 



ds= V G '^v 



d 



x'^' = — 



ds„ 



V J/ G 'W / 



(1 '^XA dv 

 i/ G ''v / dsu 



G 'W 



1 p^ /diG\n 



— I hi I I X, from (6), (4) and (8) 



G LG V du / J ' 



x'^' = o(u) X, 



Soy'"' = ©(u) y, 



z"^=: o(u) Z, 



1 1 



— — — p-<l>{1^) — 



T 



i^G 



X, Y, Z, 



'^X, '^Y, <^Z, 

 (W iS\ Ay 



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 tlirough 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 



r X = u . cos V 



-; V ^ u . sin v 



U ^ f (V) 



