TRANSFORMATION OF SURFACES BY BENDING. 



459 



facets in the same plane, and may therefore be considered as a polyhedron with plane quadri- 

 lateral facets all whose solid angles are formed by four of these facets meeting in a point. 



When the number of curves in each system is increased and their distance diminished 

 indefinitely, the plane facets of the polyhedron will ultimately coincide with the curved 

 surface, and the polygons formed by the successive edges between the facets, will coincide 

 with the lines of bending. 



These quadrilaterals may then be considered as parallelograms, the length of which is 

 determined by the portion of a curve of the second system intercepted between two curves of 

 the first, while the breadth is the distance of two ourves of the second system measured along a 

 curve of the first. The expressions for these quantities will be given when we come to the 

 calculation of our results along with the other particulars which we only specify at present. 



The angle of the sides of these parallelograms will be ultimately the same as the angle 

 of intersection of the first and second systems, which we may call <p; but if we suppose 

 the dimensions of the facets to be small quantities of the first order, the angles of the four 

 facets which meet in a point will differ from the angle of intersection of the curves at that 

 point by small angles of the first order depending on the tangential curvature of the lines 

 of bending. The sum of these four angles will differ from four right angles by a small 

 angle of the second order, the circular measure of which expresses the entire curvature of 

 the solid angle as in Art. (2). 



The angle of inclination of two adjacent facets will depend on the normal curvature of 

 the lines of bending, and will be that of the projection of two consecutive sides of the polygon 

 of one system on a plane perpendicular to a side of the other system. 



11. Explanation of the Notation to be em- 

 ployed in calculation. 



Suppose each system of lines of bending to be 

 determined by an equation containing one arbitrary 

 parameter. 



Let this parameter be u for the first system, and 

 u for the second. 



Let two curves, one from each system.be selected as 

 curves of reference,and let their parameters be w andw' . 



Let ON and OM in the figure represent these two curves. 



Let PM be any curve of the first system whose parameter is «, and PN any curve of the 

 second whose parameter is u\ then their intersection P may be defined as the point (u, u'), and 

 all quantities referring to the point P may be expressed as functions of u and u. 



Let PN, the length of a curve of the second system (u) from N, (« ) to P, (w), be 

 expressed by s, and PM the length of the curve (w) from (m' ) to («'), by s, then s and s 

 will be functions of u and u. 



Let (u + Su) be the parameter of the curve QV of the first system consecutive to PM. 

 Then the length of PQ, the part of the curve of the second system intercepted between the 

 curves (u) and (« + Sw), will be 



ds . 



du 



on. 



