AND DIAGRAMS OP FOKCES. 173 



intersecting these, and a third system passing diagonally through the intersec- 

 tions 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 corre- 

 sponding 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*; 2nd, That the stresses in the surface 

 depend entirely on the external applied forcesf ; 3rd, 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- 

 faces are not only rigid, but are capable of internal stress, independent of 

 external forces, and the expression of this stress depends on six independent 

 variables. 



* This has been shewn by Professor Jellett, Trans. R.I. A., Vol. XXH. p. 377. 

 t On the Equilibrium of a Spherical Envelope, by J. C. Maxwell. Quarterly Journal of Mathe- 

 matics, 1867. [VoL ii. p. 86.] 



