470 Mr MAXWELL, ON THE TRANSFORMATION OF SURFACES BY BENDING. 



NOTE. 



As an example of the application of the more general theory of " lines of bending," let us con- 

 sider the problem which has been already solved by Professor Jeixett. 



To determine the conditions under which one portion of a surface may be rendered rigid, while the 

 remainder is flexible. 



Suppose the lines of bending to be traced on the surface, and the corresponding polyhedron to 

 be formed, as in (9) and (10), then if the angle of one of the four edges which meet at any solid 

 angle of the polyhedron be altered by bending, those of ths other three must be also altered. These 

 edges terminate in other solid anglos, the forms of which will also be changed, and therefore the 

 effect of the alteration of one angle of the polyhedron will be communicated to every other angle 

 within the system of lines of bending which defines the form of the polyhedron. 



If any portion of the surface remains unaltered it must lie beyond the limits of the system of 

 lines of bending. We must therefore investigate the conditions of such a system being bounded. 



The boundary of any system of lines on a surface is the curve formed by the ultimate inter- 

 sections of those lines, and therefore at any given point coincides in direction with the curve of the 

 system which passes through that point. In this case there are two systems of lines of bending, 

 which are necessarily coincident in extent, and must therefore have the same boundary. At any 

 point of this boundary therefore the directions of the lines of bending of the first and second sys- 

 tems are coincident. 



But, by (7), these two directions must be " conjugate " to each other, that is, must correspond to 

 conjugate diameters of the "Conic of Contact." Now the only case in which conjugate 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 therefore by (3) be infinite at every point 

 of the curve. This is the geometrical property of what Professor Jellett 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. (453), 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 <p = 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, the surface on the 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 pos- 

 sible. The case in which the rectilinear sections form an infinite series has been discussed in Sect. (I.). 



