RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 11 



If all the faces of the polyhedron are triangles, m = 3, and we have 



p = 6(1 — n) . 



If n = 1, or in the case of a simply connected polyhedron with triangular 

 faces, p = o, that is to say, such a figure is a rigid system, which would be no 

 longer rigid if any one of its lines were wanting. In such a figure, if made of 

 material rods forming a closed web of triangles, the tensions and pressures in the 

 rods would be completely determined by the external forces applied to the figure, 

 and if there were no external force, there would be no stress in the rods. 



In a closed surface of any kind, if we cover the surface""" with a system of 

 curves which do not intersect each other, and if we draw another system inter- 

 secting these, and a third system passing diagonally through the intersections of 

 the other two, the whole surface will be covered with small curvilinear triangles, 

 and if we now substitute for the surface a system of rectilinear triangles having 

 the same angular points, we shall have a polyhedron with triangular faces 

 differing infinitely little from the surface, and such that the length of any line 

 on the surface differs infinitely little from that of the corresponding line on the 

 polyhedron. We may, therefore, in all questions about the transformation of 

 surfaces by bending, substitute for them such polyhedra with triangular faces. 



We thus find with respect to a simply connected closed inextensible surface 

 — 1st, That it is of invariable form ;t 2d, That the stresses in the surface depend 

 entirely on the external applied forces ;J %d, That if there is no external force, 

 there is no stress in the surface. 



In the limiting case of the curved surface, however, a kind of deformation is 

 possible, which is not possible in the case of the polyhedron. Let us suppose 

 that in some way a dimple has been formed on a convexo-convex part of the 

 surface, so that the edge of the dimple is a plane closed curve, and the dimpled 

 part is the reflexion in this plane of the original form of the surface. Then the 

 length of any line drawn on the surface will remain unchanged. 



Now let the dimple be gradually enlarged, so that its edge continually 

 changes its position. Every line on the surface will still remain of the same 

 length during the whole process, so that the process is possible in the case of 

 an inextensible surface. In this way such a surface may be gradually turned 

 outside in, and since the dimple may be formed from a mere point, a pressure 

 applied at a single point on the outside of an inextensible surface will not be 

 resisted, but will form a dimple which will increase till one part of the surface 

 comes in contact with another. 



In the case of closed surfaces doubly connected, p = — 6, that is, such sur- 



* On the Bending of Surfaces, by J. Clerk Maxwell. Cambridge Transactions, 1856. 

 (" This has been shown by Professor Jellett, Trans. R.I.A., vol. xxii. p. 377. 

 \ On the Equilibrium of a Spherical Envelope, by J. C. Maxwell. Quarterly Journal of 

 Mathematics, 1867. 



