24 MR CLERK MAXWELL ON 



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 corres- 

 ponding 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, we 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 exjDressing 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 (18). 



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 F and c£ for 

 <f>. We have then for the equations of relation between the two diagrams 



_ dz dz 



? ~" dx dy 



X ~ C d£ ]J ~ % 



x % + yv = cz + c K ■ 



These equations are equivalent to the following definitions : — 

 Let z in the first diagram be given as a function of x and y, z will lie on a 

 surface of some kind. Let x Q , y be particular values of x and y, and let z be the 

 corresponding value of z. Draw a tangent plane to the surface at the point 

 x , y , z , and from the point £ = 0, -q = 0, £ — — c ; in the second diagram draw 

 a normal to this tangent plane. It will cut the plane £ = at the point £ r) cor- 

 responding to xy, and the value of £ is equal and opposite to the segment of the 



