TRANSFORMATION OF SURFACES BY BENDING. 83 



For, PQ and RS being both perpendicular to Bb, RS may be turned 

 about Bb till it is parallel to PQ, in which case S< becomes = 0. 



By repeating this process, we may make all the " shortest lines" parallel to 

 one another, and then all the generating lines will be parallel to the same 

 plane. 



We have hitherto considered generating lines situated at finite distances from 

 one another ; but what we have proved will be equally true when their distances 

 are indefinitely diminished. Then in the limit 



becomes -j- , 



U t tt, 



W d0_ 



u t tt, du ' 



Scr dcr 



u M du ' 



d(f> 



du ' 



All these quantities being functions of u, , 0, cr and <}>, are functions of u 

 and of each other ; and if the forms of these functions be known, the positions 

 of all the generating lines may be successively determined, and the equation 

 to the surface may be found by integrating the equations containing the values 

 of , 0, a- and <f>. 



When the surface is bent in any manner about the generating lines, , 0, 

 and cr remain unaltered, but (j> is changed at every point. 



The form of <f> as a function of u will depend on the nature of the 

 bending ; but since this is perfectly arbitrary, < may be any arbitrary function 

 of u. In this way we may find the form of any surface produced by bending 

 the given surface along its generating lines. 



By making <j> = 0, we make all the generating lines parallel to the same 

 plane. Let this plane be that of xy, and let the first generating line coincide 

 with the axis of x, then will be the height of any other generating line 

 above the plane of xy, and the angle which its projection on that plane 

 makes with the axis of x. The ultimate intersections of the projections of the 

 generating lines on the plane of xy will form a curve, whose length, measured 

 from the axis of x, will be cr. 



112 



