RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 29 



/(*! + *D* <27) ' 



These line integrals are to be taken round the closed curve s. If we take a 

 point in the curve s as origin in the original body, and the corresponding point 

 in cr as origin in the diagram of stress, then £ and 77 are the components of the 

 whole stress on the right hand of the curve from the origin to a given point. 

 If p denote the line joining the origin with the point £77, then p will represent in 

 direction and magnitude the whole stress on the arc <x. 



The line integral may now be interpreted as the work done on a point which 

 travels once round the closed curve s, and is everywhere acted on by a force 

 represented in direction and magnitude by p. We may express this quantity 

 in terms of the stress at every point of the curve, instead of the resultant stress 

 on the whole arc, as follows : — 



For integrating (27) by parts it becomes, 



"/ (' S + ,J t) ds = - /< x * + Y ?<) * ^ 



or if Hds 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 



ff '(Pj + T 2 )dxdy = -J 'Br cos eds (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. 13, 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 equili- 

 brium. 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 multi- 

 plied by the section, so that the integral Jf(Pi + P 2 ) dxdy 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 direc- 

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

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

 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. 



J J \dx dy dy dxj J ' 

 VOL. XXVI. PART I. H 



