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system is quite independent of that round any other ; but in those 

 which follow, the bending at one point influences that at every other 

 point. The case of a surface of revolution bent symmetrically with 

 respect to the axis is taken as an example. 



The remainder of the paper contains an elementary investigation 

 of the conditions of bending of a surface of any form. 



The surface is considered as the limit of the inscribed polyhedron 

 when the number of the sides is increased and their size diminished 

 indefinitely. 



A method is then given by which a polyhedron with triangular 

 facets may be inscribed in any surface ; and it is shown, that when 

 a certain condition is fulfilled, the triangles unite in pairs so as to 

 form a polyhedron with quadrilateral facets. The edges of this 

 polyhedron form two intersecting systems of polygons, which become 

 in the limit curves of double curvature ; and when the condition 

 referred to is satisfied, the two systems of curves are said to be 

 " conjugate " to one another. 



The solid angle formed by four facets which meet in a point is 

 then considered, and in this way a " measure of curvature " of the 

 surface at that point is obtained. 



It is then shown that if there be two surfaces, one of which has 

 been developed from the other, one, and only one, pair of systems of 

 corresponding lines can be drawn on the two surfaces so as to be 

 conjugate to each other on both surfaces. This pair of systems 

 completely determines the nature of the transformation, and is called 

 a double system of "lines of bending." By means of these lines 

 the most general cases are reduced to that of the quadrilateral poly- 

 hedron. The condition to be fulfilled at every point of the surface 

 during bending is deduced from the consideration of one solid angle 

 of the polyhedron. It is found that the product of the principal 

 radii of curvature is constant. 



By considering the angles of the four edges which meet in a point, 

 we obtain certain conditions, which must be satisfied by the lines of 

 bending in order that any bending may be possible. If one of these 

 conditions be satisfied, an infinitesimal amount of bending may take 

 place, after which the system of lines must be altered that the bend- 

 ing may continue. Such lines of bending are in continual motion 

 over the surface during bending, and may be called " instantaneous 

 lines of bending." When a second condition is satisfied, a finite 

 amount of bending may take place about the same system of lines. 

 Such a system may be called a " permanent system of lines of 

 bending." 



Every conception required by the problem is thus rendered per- 

 fectly definite and intelligible, and the difficulties of further investi- 

 gation are entirely analytical. No attempt has been made to over- 

 come these, as the elementary considerations previously employed 

 would soon become too complicated to be of any use. 



For the analytical treatment of the subject the reader is referred 

 to the following memoirs : — 



1. " Disquisitiones generales circa superficies curvas," by M. C. 



