194 



RECIPROCAL FIGURES, FRAMES, 



Hence 



tan -a 



_ 



da* dy' 

 cPF 



(24). 



Consider the area bounded by a closed curve s, and let us determine the sur- 

 face integral of the sum of the principal stresses over the area within the curve. 



The integral is 



By a well-known theorem, corresponding in two dimensions to that of Green in 

 three dimensions, the latter expression becomes, when once integrated, 



( (dF dx _dFdy\ . . 



}\dy ds dxds) a 



or 



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 <r as origin in the diagram of stress, then and 17 are the compo- 

 nents 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 17, then 

 p will represent in direction and magnitude the whole stress on the arc o: 



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

 which travels once round the closed curve *, 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, 



,(28), 



