TRANSFORMATION OF SURFACES BY BENDING. 113 



This, however, is true only for one half of the surface when bent. The 

 other half is precisely symmetrical, but belongs to a surface which is not con- 

 tinuous with the first. 



The surface in its original form is divided by the plane of xy into two 

 parts which meet in that plane, forming a kind of cuspidal edge of a circular 

 form which limits the possible value of s and r. 



After being bent, the surface still consists of the same two parts, but the 

 edge in which they meet is no longer of the cuspidal form, but has a finite 



angle = 2 cos" 1 - , and the two sheets of the surface become parts of two different 

 P 



surfaces which meet but are not continuous. 



NOTE. 



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

 consider the problem which has been already solved by Professor Jellett. 



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

 tiie remainder is flexible. 



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

 hedron 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 the other three 

 must be also altered. These edges terminate in other solid angles, 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- 

 section 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 systems are coincident. 



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

 spond to conjugate diameters of the " Conic of Contact." Now the only case in which con- 



VOL. I. 15 



