TRANSFORMATION OF SURFACES BY BENDING. 447 



Now suppose the part of the surface between the lines Aa and Bb to be fixed, while the 

 part between Bb and Cc is turned round Bb as an axis. The line RS will then revolve 

 round the point R, remaining perpendicular to Bb, and Cc will still be at the same distance 

 from Bb, and will make the same angle with it. Hence of the four quantities $<£>, $9 2 , 8<r 

 and Scf), fob alone will be changed by the process of bending. 8<p, however, may be varied in a 

 perfectly arbitrary manner, and may even be made to vanish. 



For, PQ and RS being both perpendicular to Bb, RS may be turned about Bb till it is 

 parallel to PQ, in which case $(p 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 



u 2 — u x du 



All these quantities being functions of u, £, 9, <y, and (p, 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 £, 9, a, and d>. 



When the surface is bent in any manner about the generating lines, £, 9, and a remain 

 unaltered, but d> is changed at every point. 



The form of d) as a function of m will depend on the nature of the bending ; but since this 

 is perfectly arbitrary, d> 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 = 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 X, w iU 

 be the height of any other generating line above the plane of xy, and 9 the angle which its 

 projection on that plane makes with the axis of x. The ultimate intersections of the pro- 

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

 from the axis of x, will be <x. 



Since in this case the quantities X, 9, and a are represented by distinct geometrical 



quantities, we may simplify the consideration of all surfaces generated by straight lines by, 



reducing them by bending to the case in which those lines are parallel to a given plane. 



dX 

 In the class of surfaces in which the generating lines ultimately intersect, — ' * = 0, and 



Vol. IX. Part IV. 58 



