100 TRANSFORMATION OF SURFACES BY BENDING. 



Let ON and OM in the figure represent these two curves. 



Let PM be any curve of the first system whose parameter is u, and PN 

 any curve of the second whose parameter is u', then their intersection P may 

 be defined as the point (u, u'), and all quantities referring to the point P may 

 be expressed as functions of u and u'. 



Let PN, the length of a curve of the second system (u), from N (w ) to P 

 (u), be expressed by s, and PM the length of the curve (u) from (',) to (u), by 

 /, then s and s' will be functions of u and u. 



Let (u + 8u) be the parameter of the curve QV of the first system consecu- 

 tive to PM. Then the length of PQ, the part of the curve of the second system 

 intercepted between the curves (u) and (u + 8u), will be 



8u 

 du 



Similarly PR may be expressed by 



8u' 



These values of PQ and PR will be the ultimate values of the length and 

 breadth of a quadrilateral facet. 



The angle between these lines will be ultimately equal to <}>, the angle of 

 intersection of the system ; but when the values of Stt and 8u' are considered as 

 finite though small, the angles a, b, c, d of the facets which form a solid angle 

 will depend on the tangential curvature of the two systems of lines. 



Let r be the tangential curvature of a curve of the first system at the 

 given point measured in the direction in which u increases, and let ?', that of the 

 second system, be measured in the direction in which u' increases. 



Then we shall have for the values of the four plane angles which meet at P, 



1 ds 1 ds 



I _, 1 ds ^ , I ds ~ 



, 1 <&' s , !_ * s 



, , 1 ds' _ , 1 ds ^ 



