TRANSFORMATION OF SURFACES BY BENDING. 



considered as a polyhedron with plane quadrilateral 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 inter- 

 cepted between two curves of the first, while the breadth is the distance of 

 two curves 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 <f> ; 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 con- 

 secutive 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 employed in calculation. 



Suppose each system of lines of bend- 

 ing to be determined by an equation con- 

 taining 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 u, and u\. 



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