TRANSFORMATION OF SURFACES BY BENDING. 93 



invariable, the bending of the polyhedron cannot alter the measure of curvature 

 of each of its solid angles. 



If perpendiculars be drawn to the sides of the polyhedron which contain 

 other solid angles, additional points on the sphere will be found, and if these 

 be joined by arcs of great circles, a network of hexagons will be formed on 

 the sphere, each of which corresponds to a solid angle of the polyhedron and 

 represents its " entire curvature." 



The entire curvature of any assigned portion of the polyhedron is the sum 

 of the entire curvatures of the solid angles it contains. It is therefore repre- 

 sented by a polygon on the sphere, which is composed of all the hexagons 

 corresponding to its solid angles. 



If a polygon composed of the edges of the polyhedron be taken as the 

 boundary of the assigned portion, the sum of its exterior angles will be the 

 same as the sum of the exterior angles of the polygon oil the sphere ; but 

 the area of a spherical polygon is equal to the defect of the sum of its 

 exterior angles from four right angles, and this is the measure of entire curva- 

 ture. 



Therefore the entire curvature of the portion of the polyhedron enclosed 

 by the polygon is equal to the defect of the sum of its exterior angles from 

 four right angles. 



Since the entire curvature of each solid angle is unaltered by bending, 

 that of a finite portion of the surface must be also invariable. 



3. On the " Conic of Contact," and its use in determining the curvature 

 of normal sections of a surface. 



Suppose the plane of one of the triangular facets of the polyhedron to 

 be produced till it cuts the surface. The form of the curve of intersection 

 will depend on the nature of the surface, and when the size of the triangle 

 is indefinitely diminished, it will approximate to the form of a conic section. 



For we may suppose a surface of the second order constructed so as to 

 have a contact of the second order with the given surface at a point within 

 the angular points of the triangle. The curve of intersection with this surface 

 will be the conic section to which the other curve of intersection approaches. 

 This curve will be henceforth called the " Conic of Contact," for want of a better 

 name. 



