Cambridge Philosophical Society. 449 



mean motions of the Apse and Node. See the former paper, Phil. 

 Mag. vol. vii. p. 278. 



Also a paper by Mr. J. Clerk Maxwell on the Transformation -tof ■' 

 Surfaces by Bending. 



The kind of transformation here considered is that in which a 

 surface changes its form without extension or contraction of any of 

 its parts. Such a process may be called bending or development. 

 The most obvious case is that in which the surface is originally a 

 plane, and becomes, by bending, one of the class called " developable 

 surfaces." Surfaces generated by straight lines, which do not ulti- 

 mately intersect, may also be bent about these straight lines as axes. 

 In this way they may be transformed into surfaces whose generating 

 lines are parallel to a given plane, just as the former class are trans- 

 formed into planes. 



In both these cases, the bending round one straight line of the 

 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 



Phil. Mag, S. 4. Vol. 7. No. 47. June 1854. 3 H 



