RECIPROCAL FIGURES, FRAMES, 



If p represents in direction and magnitude the resultant of the stress on 

 the curve * from the origin to a point which moves round the curve, then the 

 area traced out by p is equal to the surface- integral required. If Xds and ) "</.< 

 are the components of the stress on the element ds, and I the whole length of 

 the closed curve *, then the surface integral is equal to either of the quantities 



P Y ('Xds .ds, or-fx (' Yds . ds. 



In a frame the stress in each piece is entirely longitudinal, so that the 

 product of the principal stresses is zero, and therefore nothing is contributed to 

 the surface-integral except at the points where the pieces meet or cross each 

 other. To find the value of the integral for any one of these points, draw a 

 closed curve surrounding it and no other point, and therefore cutting all the 

 pieces which meet in that point in order. The corresponding figure in the 

 diagram of stress will be a polygon, whose sides represent in magnitude and 

 direction the tensions in the several pieces taken in order. The area of this 

 j>olygon, therefore, represents the value of ^P 1 P t dxdy for the point of concourse, 

 and is to be considered positive or negative, according as the tracing point 

 travels round it in the positive or the negative cyclical direction. 



Hence the following theorem, which is applicable to all plane frames, whether 

 a diagram of forces can be drawn or not. 



For each point of concourse or of intersection construct a polygon, liy 

 drawing in succession lines parallel and proportional to the forces acting on the 

 point in the several pieces which meet in that point, taking the pieces in 

 cyclical order round the point. The area of this polygon is to be taken positive 

 or negative, according as it lies on the left or the right of the tracing point. 



If, then, a closed curve be drawn surrounding the entire frame, and a 

 polygon be drawn by drawing in succession lines parallel and proportional to 

 all the external forces which act on the frame in the order in which their 

 lines of direction meet the closed curve, then the area of this polygon is equal 

 to the algebraic sum of the areas of the polygons corresponding to the various 

 joints of the frame. 



In this theorem a polygon is to be drawn for every point, whether the 

 lines of the frame meet or intersect, whether they are really jointed together, 





