TRANSFORMATION OF SURFACES BY BENDING. 



461 



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 0, and a and c to (ir-(p), 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 = It sin (p. 



By the expressions for I and X in the last article, we find 



7 \ ds ds ' s 5 ' 



k = — -. — j ; — , cu du 



pp sin <p du du 

 for the entire curvature of one solid angle. 



Since the whole number of solid angles is equal to the whole number of facets, we may 

 suppose a quarter of each of the facets of which it is composed to be assigned to each solid 

 angle. The area of these will be the same as that of one whole facet, namely, 



. ds ds . . , 



sin d> — — ■. Su Su' ; 

 ' du du 



therefore dividing the expression for k by this quantity, we find for the value of the specific 

 curvature at P 



1 

 pp sin 3 <p 

 which gives the specific curvature in terms of the normal curvatures of the lines of bending 

 and their angle of intersection. 



13. Farther reduction of this expression by means of the " Conic of Contact" as defined 

 in Art. (3). 



Let a and b be the semiaxes of the conic of contact, and h the sagitta or perpendicular to 

 its plane from the centre to the surface. 



Let CP, CQ be semidiameters parallel to the lines of bending 

 of the first and second systems, and therefore conjugate to 

 each other. 



By (Art. 3,) 



and p 

 and the expression for p in Art. (12), becomes 



1 CP 2 



1 CQ^ 



2 h ' 



P = 



4A 2 



(CP.CQ sin 0) 2 



But CP.CQ sin is the area of the parallelogram CPRQ, which is one quarter of the 

 circumscribed parallelogram, and therefore by a well-known theorem 



CP.CQ sin (p = ab, 



