TRANSFORMATION OF SURFACES BY BENDING. 101 



These values are correct as far as the first order of small quantities. Those 

 corrections which depend on the curvature of the surface are of the second order. 



Let p be the normal curvature of a curve of the first system, and p' that 

 of a curve of the second, then the inclination I of the plane facets a and b, 

 separated by a curve of the second system, will be 



ds' 



' I 7 / Vw 



p sin <p du 

 as far as the first order of small angles, and the inclination I' of b and c will be 



7 , 1 ds * 

 I =-7. 1 -^-8u 

 p siu<j> du 



to the same order of exactness. 



12. On the corresponding polygon on the surface of the sphere of reference. 



By the method described in Art. (2) we may 

 find a point on the sphere corresponding to each 

 facet of the polyhedron. 



In the annexed figure, let a, b, c, d be the 

 points on the sphere corresponding to the four facets 

 which meet at the solid angle P. Then the area 

 of the spherical quadrilateral a, b, c, d will be the 

 measure of the entire curvature of the solid angle P. 



This area is measured by the defect of the sum of the exterior angles 

 from four right angles ; but these exterior angles are equal to the four angles 

 a, b, c, d, which form the solid angle P, therefore the entire curvature is 

 measured by 



Since a, b, c, d are invariable, it is evident, as in Art. (2), that the entire 

 curvature at P is not altered by bending. 



By the last article it appears that when the facets are small the angles b 

 and d are approximately equal to $, and a and c to (IT <j>), and since the sides 

 of the quadrilateral on the sphere are small, we may regard it as approximately 

 a plane parallelogram whose angle bad = <f>. 



The sides of this parallelogram will be I and I', the supplements of the 

 angles of the edges of the polyhedron, and we may therefore express its area 

 as a plane parallelogram 



k = IV sin <f>. 



