RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 27 



point within it. Let us take a and b two contiguous points in different cells, 

 then a and /3 will be the points at a finite distance to which these cells are 



a 



reciprocal, and A 2 F ="-r > which becomes infinite when ab vanishes. 



If a and b are in the surface bounding the cells, a and /3 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, depend- 

 ing 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 equilibrium 

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



S*- + ajf** = o and &r« + -%p» = ° • ■ < 19 )> 



whence it follows that 



*-=W Pxv== ~^dy *» = <& • ■ • (20) > 



where F is a function of x and y, the form of which is (as far as these equations 

 are concerned) perfectly arbitrary, and the value of which at any point is in- 

 dependent of the choice of axes of co-ordinates. Since the stresses depend on 

 the second derivatives of F, any linear function of x and y may be added to F 

 without affecting the value of the stresses deduced from F. Also, since the 

 stresses are linear functions of F, any two systems of stress may be mechanically 

 compounded by adding the corresponding values of F. 



The importance of Airy's function in the theory of stress becomes even more 



* Phil. Trans. 1863. 



