EQUILIBRIUM OF AN ISOTROPIO ELASTIC ROD OF CIRCULAR SECTION. 973 



Hence, with the help of Green's Theorem, which gives I ^c?S= I ^iv-^ds, etc.. 



2X"' = - -IttE^'^-^ , 2Y'" = - IttE^^^ 

 4 dz^ 4 dz'^ 



(36) 



fZ22 ' ^N. , / 2 dz^ 



I 



Here for the first time in the process we have an undetermined function of z for 

 each of the three resultants and for the ^»; couple. 

 Two alternatives now present themselves, 

 (i) The given applied forces may come in at this point. 

 The conditions of integrability (14) and (15) then give 



2X - i-7rE^5t^ = 0> 2Y - J-,rE^^2 = 0, \ 



'- (37) 



SZ + .E'^o.o, 2(|Y-,X) + l.,2^o.o.j 



(ii) The given applied forces may not come in till later. 

 In this case 



^^U-2 _o ^^^-2_o dm,_^ 'i'^-.Q. 



dz^ ' dz^ ' dz^ ' dz^ 

 the forces in (3.5) will not involve U_2, V_2, Wq, Oo, nor will the corresponding forces 

 in the equations for u^, v^, iv^. 



The terms already found involving these functions are therefore exact solutions 

 under no body force or traction at the cylindrical surface, as of course we know 

 already. We thus incidentally fall upon the permanent free modes of equilibrium 

 of the cylinder by a distinctly more analytical process than the semi-inverse method 

 by which St-Venant originally discovered them. 



The choice between the alternatives (i) and (ii) is controlled by the conditions 

 assigned at the plane ends. 



But there is practically no loss of generality in fixing at once on the alternative (i) 

 as we shall do here. In fact, it is easy to see that (ii) leads to the same solution as 

 (i), with the addition merely of undetermined permanent modes. 



Seeing that the integrations in (37) are taken over the section of the cylinder 

 ^^ + >f= 1 and the circle bounding that section, it is obviously implied that the forces 

 X, Y, Z are of order zero in a. 



Since the equations of plane strain, and Laplace's equation, are easily solved for 

 a circular boundary, we could now find u^, v^, iv^ completely. And it is easy to see 

 that the above process can be continued indefinitely, assuming only that the 

 0-difi"erentiations requisite for calculating the forces at each step can be performed. 

 For obviously at the next step U_i, V_i, W^, and 12i will be available for the purpose of 

 satisfying the conditions of integrability of the equations for Wg, v^, w^ ; and 



- r\ ■■ "\ ^V, ^-^will be given by these conditions ; similarly at the next step 

 (X2* dz dz dz" 



with Uo, Vo, W2, and O2, and so on indefinitely. 



