482 X. STATICS OF DEFORMABLE BODIES. 



Put 



and we have 



u = rj lax + j 



91) 



Integrating 84), 



92) w = (a + a t x + a,y) z + ^ (6 + \x + 6 a y) + F(x, y), 

 which inserted in e) gives 



If we put 



94) F = & - (b ^^- 2 + \xf + b 2 x*y) -Vx- Wy 

 we get finally 



or 5i is a harmonic function of the variables x, y. The conditions 71) 

 at the origin give 



a 1 = a" 



We have now for the traction, 

 97) Z.-1. + *_* 



= j&fa + a^ 4- a^y + z(b + 6^ 4- 



If we take for the origin the center of mass of the cross -section, 

 so that 



C CxdS = C CydS = 0, 



we have, integrating over the cross -section, z = const., 

 98) C Cz z dS = E C C(a -f Iz) dS =E(a + be) S. 



