30 me clerk maxwell on 



or by transformation of variables 



-ff, m 



Hence the surface integral of the product of the principal stresses within 

 the curve is equal to the area of the corresponding curve o- in the diagram of 

 stress, and therefore depends entirely on the external stress on the curve .«. 

 This is seen from the construction of the curve <r in the diagram of stress, since 

 each element da represents the stress on the corresponding element ds of the 

 original curve. 



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

 curve s 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 Yds 

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

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



/ T fXds . ds , or - f X fxds . ds . 



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

 duct 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 

 polygon, therefore, represents the value of 'f/V \P 2 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, by draw- 

 ing 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 poly- 

 gon 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 points 

 of the frame. 



