98 TRANSFORMATION OF SURFACES BY BENDING. 



The surface will then be covered with a system of quadrilaterals, the size 

 of which may be diminished indefinitely by interpolating values of u and u 

 between those already assumed; and in the limit each quadrilateral may be 

 regarded as a parallelogram coinciding with a facet of the inscribed polyhedron. 



The length, the breadth, and the angle of these parallelograms will vary at 

 different parts of the surface, and will therefore depend on the values of 

 and u'. 



The curvature of a line drawn on a surface may be investigated by consider- 

 ing the curvature of two other lines depending on it. 



The first is the projection of the line on a tangent plane to the surface at 

 a given point in the line. The curvature of the projection at the point of 

 contact may be called the tangential curvature of the line on the surface. It 

 has also been called the geodesic curvature, because it is the measure of its 

 deviation from a geodesic or shortest line on the surface. 



The other projection necessary to define the curvature of a line on the 

 surface is on a plane passing through the tangent to the curve and the normal 

 to the surface at the point of contact. The curvature of this projection at that 

 point may be called the normal curvature of the line on the surface. 



It is easy to shew that this normal curvature is the same as the curvature 

 of a normal section of the surface passing through a tangent to the curve at 

 the same point. 



10. General considerations applicable to tlie inscribed polyhedron. 



When two series of lines of bending belonging to the first and second systems 

 have been described on the surface, we may proceed, as in Art. (1), to describe 

 a third series of curves so as to pass through all their intersections and form 

 the diagonals of the quadrilaterals formed by the first pair of systems. 



Plane triangles may then be constituted within the surface, having these 

 points of intersection for angles, and the size of the facets of this polyhedron may 

 be diminished indefinitely by increasing the number of curves in each series. 



But by Art. (8) the first and second systems of lines of bending are conju- 

 gate to each other, and therefore by Art. (4) the polygon just constructed will 

 have every pair of triangular facets in the same plane, and may therefore be 



