AND DIAGRAMS OF FORCES. 193 



where F is a function of x and y, the form of which is (as far as these 

 equations are concerned) perfectly arbitrary, and the value of which at any 

 point is independent of the choice of axes of co-ordinates. Since the stresses 

 depend on the second derivatives of F, any linear function of x and y may be 

 added to F without affecting the value of the stresses deduced from F. Also, 

 since the stresses are linear functions of F, any two systems of stress may be 

 mechanically compounded by adding the corresponding values of F. 



The importance of Airy's function in the theory of stress becomes even 

 more manifest when we deduce from it the diagram of stress, the co-ordinates 

 of whose points are 



dF dF 



For if s be the length of any curve in the original figure, and cr that of the 

 corcesponding curve in the diagram of stress, and if Xds, Yds are the com- 

 ponents of the whole stress acting on the element ds towards the right hand 

 of the curve s 



dy , d'F dy 1 dg dy , dg , 

 Xds = p xx ~r-ds= -y-; -f-ds = -r L -f ds = -f- da- 

 em ds dy 2 ds dy ds da- 



dx , d s F dx , d-n dx , d-n 

 and ---^ 



Hence the stress on the right hand side of the element ds of the original 

 curve is represented, both in direction and magnitude, by the corresponding 

 element da- of the curve in the diagram of stress, and, by composition, the 

 resultant stress on any finite arc of the first curve s is represented in direction 

 and magnitude by the straight line drawn from the beginning to the end of 

 the corresponding curve <r. 



If P v P a are the principal stresses at any point, and if P, is inclined a 

 to the axis of x, then the component stresses are 



jp^ = P, cos 3 a + P 3 sin 2 a i 



^^(P.-P^sinacosa .......................... (23). 



p m = P l sin 1 a + P s cos 2 a J 



VOL. II. 25 



