114 TRANSFORMATION OF SURFACES BY BENDING. 



jugate diameters of a conic can coincide, is when the conic is an hyperbola, and both diameters 

 coincide with one of the asymptotes ; therefore the boundary of the system of lines of bending 

 must be a curve at every point of which the conic of contact is an hyperbola, one of whose 

 asymptotes lies in the direction of the curve. The radius of " normal curvature " must there- 

 fore by (3) be infinite at every point of the curve. This is the geometrical property of 

 what Professor Jcllett calls a " Curve of Flexure," so that we may express the result as 

 follows : 



If one portion of a surface be fixed, while the remainder is bent, the boundary of the fixed 

 portion is a curve of flexure. 



This theorem includes those given at p. (92), relative to a fixed curve on a surface, for in 

 a surface whose curvatures are of the same sign, there can be no " curves of flexure," and 

 in a developable surface, they are the rectilinear sections. Although the cuspidal edge, or 

 arete de rebroussement, satisfies the analytical condition of a curve of flexure, yet, since its 

 form determines that of the whole surface, it cannot remain fixed while the form of the surface 

 is changed. 



In concavo-convex surfaces, the curves of flexure must either have tangential curvature or 

 be straight lines. Now if we put ^=0 in the equations of Art. (17), we find that the 

 lines of bending of both systems have no tangential curvature at the point where they touch 

 the curve of flexure. They must therefore lie entirely on the convex side of that curve, and 

 therefore 



If a curve of flexure be fixed, Hie surface on tiie concave side of the curve is not flexible. 



I have not yet been able to determine whether the surface is inflexible on the convex side 

 of the curve. It certainly is so in some cases which I have been able to work out, but I 

 have no general proof. 



When a surface has one or more rectilinear sections, the portions of the surface between 

 them may revolve as rigid bodies round those lines as axes in any manner, but no other motion 

 is possible. The case in which the rectilinear sections form an infinite series has been discussed 

 in Sect. (I.). 



