192 RECIPROCAL FIGURES, FRAMES, 



Since /' in each cell is a linear function of x, y, z, there can be no stress 

 at any point within it. Let us take a and 6 two contiguous points in different 

 oells, then a and ft will be the points at a finite distance to which these cells 



are reciprocal, and A*.F=^j, which becomes infinite when ab vanishes. 



If a and b are hi the surface bounding the cells, a and ft coincide. Hence 

 there is a stress in this surface, uniform in all directions in the plane of the 

 surface, and such that the stress across unit of length drawn on the surface is 

 proportional to the distance between the points which are reciprocal to the two 

 cells bounded by the surface, and this stress is a tension or a pressure according 

 as the two points are similarly or oppositely situated to the two cells. 



The kind of equilibrium corresponding to this case is therefore that of a 

 system of liquid films, each having a tension like that of a soap bubble, 

 depending on the nature of the fluid of which it is composed. If all the films 

 are composed of the same fluid, their tensions must be equal, and all the edges 

 of the reciprocal diagram must be equal. 



On Airy's Function of Stress. 



Mr Airy, in a paper " On the Strains in the Interior of Beams*," was, I 

 believe, the first to point out that, in any body in equilibrium under the action 

 of internal stress in two dimensions, the three components of the stress in any 

 two rectangular directions are the three second derivatives, with respect to these 

 directions, of a certain function of the position of a point in the body. 



This important simplification of the theory of the equilibrium of stress in 

 two dimensions does not depend on any theory of elasticity, or on the mode 

 in which stress arises in the body, but solely on the two conditions of equi- 

 librium of an element of a body acted on only by internal stress 



*-+*-' and 



whence it follows that 



d'F 



> and A. - ............... (20)l 



* Phil. Trans. 1863. 



