188 RECIPROCAL FIGURES, FRAMES, 



the first cell. In this way, by making the positive side of the surface con- 

 tinuous throughout each cell, and by changing it when we pass to the next 

 cell, we may settle the positive and negative side of every face of every cell, 

 the sign of every face depending on which of the two cells it is considered 

 for the moment to belong to. 



If we now suppose forces of tension or pressure applied along every edge of 

 the first diagram, so that the force on each extremity of the edge is in the 

 direction of the positive normal to the corresponding face of the cell corre- 

 sponding to that extremity, and proportional to the area of the face, then these 

 pressures and tensions along the edges will keep every point of the diagram 

 in equilibrium. 



Another way of determining the nature of the force along any edge of 

 the first diagram, is as follows : 



Round any edge of the first diagram draw a closed curve, embracing it 

 and no other edge. However small the curve is, it will enter each of the 

 cells which meet in the edge. Hence the reciprocal of this closed curve will 

 be a plane polygon whose angles are the points reciprocal to these cells taken 

 in order. The area of this polygon represents, both in direction and magnitude, 

 the whole force acting through the closed curve, that is, in this case the stress 

 along the edge. If, therefore, in going round the angles of the polygon, w- 

 travel in the same direction of rotation in space as in going round the closed 

 curve,' the stress along the edge will be a pressure ; but if the direction is 

 opposite, the stress will be a tension. 



This method of expressing stresses in three dimensions comprehends all 

 cases in which Rankine's reciprocal figures are possible, and is applicable to 

 certain cases of continuous stress. That it is not applicable to all such cases 

 is easily seen by the example of p (189). 



On Reciprocal Diagrams in two Dimensions. 



If we make F a function of x and y only, all the properties already 

 deduced for figures in three dimensions will be true in two ; but we may form 

 a more distinct geometrical conception of the theory by substituting cz for /' 



