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TRANSFORMATION OF SURFACES BY BENDING. 



ON THE PROPERTIES OF A SURFACE CONSIDERED AS THE LIMIT OF THE INSCRIBED 



POLYHEDRON. 



1. To inscribe a polyhedron in a given surface, all whose sides shatt be 

 triangles, and all whose solid angles shall be hexahedral. 



On the given surface describe a series of curves 

 according to any assumed law. Describe a second series 

 intersecting these in any manner, so as to divide the 

 whole surface into quadrilaterals. Lastly, describe a 

 third series (the dotted lines in the figure), so as to 

 pass through all the intersections of the first and second 

 series, forming the diagonals of the quadrilaterals. 



The surface is now covered with a network of curvilinear triangles. The 

 plane triangles which have the same angular points will form a polyhedron 

 fulfilling the required conditions. By increasing the number of the curves in 

 each series, and diminishing their distance, we may make the polyhedron 

 approximate to the surface without limit. At the same time the polygons 

 formed by the edges of the polyhedron will approximate to the three systems 

 of intersecting curves. 



2. To find the ineasure of the " entire curvature " of a solid angle of the 

 polyhedron, and of a finite portion of its surface. 



From the centre of a sphere whose radius is unity draw perpendiculars to 

 the planes of the six sides forming the solid angle. These lines will meet the 

 surface in six points on the same side of the centre, which being joined by 

 arcs of great circles will form a hexagon on the surface of the sphere. 



The area of this hexagon represents the entire curvature of the solid angle. 



It is plain by spherical geometry that the angles of this hexagon are the 

 supplements of the six plane angles which form the solid angle, and that the 

 arcs forming the sides are the supplements of those subtended by the angles 

 of the six edges formed by adjacent sides. 



The area of the hexagon is equal to the excess of the sum of its angles 

 above eight right angles, or to the defect of the sum of the six plane angles 

 from four right angles, which is the same thing. Since these angles are 



