TRANSFORMATION OF SURFACES BY BENDING. 95 



Therefore the directions of two curves of the first and second system at 

 their point of intersection must be parallel to two conjugate diameters of the 

 conic of contact at that point in order that such a polyhedron may be inscribed. 



Systems of curves intersecting in this manner will be referred to as "conju- 

 gate systems." 



5. On the elementally conditions of the applicability of two surfaces. 



It is evident, that if one surface is capable of being applied to another by 

 bending, every point, line, or angle in the first has its corresponding point, line, 

 or angle in the second. 



If the transformation of the surface be effected without the extension or 

 contraction of any part, no line drawn on the surface can experience any change 

 in its length, and if this condition be fulfilled, there can be no extension or 

 contraction. 



Therefore the condition of bending is, that if any line /whatever be drawn 

 on the first surface, the corresponding curve on the second surface is equal to it 

 in length. All other conditions of bending may be deduced from this. 



6. If two curves on the first surface intersect, the corresponding curves on the 

 second surface intersect at the same angle. 



On the first surface draw any curve, so as to form a triangle with the 

 curves already drawn, and let the sides of this triangle be indefinitely dimin- 

 ished, by making the new curve approach to the intersection of the former 

 curves. Let the same thing be done on the second surface. We shall then 

 have two corresponding triangles whose sides are equal each to each, by (5), 

 and since their sides are indefinitely small, we may regard them as straight 

 lines. Therefore by Euclid I. 8, the angle of the first triangle formed by the 

 intersection of the two curves is equal to the corresponding angle of the second. 



7. At any given point of the first surface, two directions can be found, which 

 are conjugate to each other with respect to the conic of contact at that point, and 

 continue to be conjugate to each other when the first surface is transformed into the 

 second. 



For let the first surface be transferred, without changing its form, to a 

 position such that the given point coincides with the corresponding point of the 

 second surface, and the normal to the first surface coincides with that of the 



