AND DIAGRAMS OP FORCES. 195 



or if Rds is the actual stress on ds, and r is the radius vector of ds, and 

 if R makes with r an angle e, we obtain the result 



ds ..................... (29). 



This line integral, therefore, which depends only on the stress acting on 

 the closed curve s, is equal to the surface integral of the sum of the principal 

 stresses taken over the whole area within the curve. 



If there is no stress on the curve s acting from without, then the surface 

 integral vanishes. This is the extension to the case of continuous stress of the 

 theorem, given at p. 176, that the algebraic sum of all the tensions multiplied 

 each by the length of the piece in which it acts is zero for a system in 

 equilibrium. In the case of a frame, the stress in each piece is longitudinal, 

 and the whole pressure or tension of the piece is equal to the longitudinal 

 stress multiplied by the section, so that the integral //(Pj + PJcZxcfo/ for each 

 piece is its tension multiplied by its length. 



If the closed curve s is a small circle, the corresponding curve cr will be 

 an ellipse, and the stress on any diameter of the circle will be represented in 

 direction and magnitude by the corresponding diameter of the ellipse. Hence, 

 the principal axes of the ellipse represent in direction and magnitude the prin- 

 cipal stresses at the centre of the circle. 



Let us next consider the surface integral of the product of the principal 

 stresses at every point taken over the area within the closed curve s. 



= ( rii 2 _ ^ 11V ] ^jjj^y 



J J \rfx c% cfo/ cfay ' *" 

 or by transformation of variables 



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

 the curve is equal to the area of the corresponding curve cr in the diagram of 

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

 This is seen from the construction of the curve cr in the diagram of stress, 

 since each element do- represents the stress on the corresponding element ds of 



the original curve. 



252 



