EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 969 



~ 1S2K. "0) + ^V^^^"-ij 



+ a 



'- = 0, 



+ 



(10) 



+ a-"J^8.^{w_^) + /^g-(^M-2 + v^-2)] 

 + 



1 = 0. 



(H) 



We have written down the given forces at the head of the series ; but their 

 actual place among the other terms is as yet unknown. Obviously they cannot by 

 themselves be the lowest terms of the series, since the terms of every order must, in 

 each of the six series, separately vanish. Can they be of the same order as the next 

 terms of the series ; in other words, can p he — 4 ? This would imply that u_2, v_2, w^^ 

 are given by 



Di(m_2 , ^^-2) + X = 0, ) 8i(w_2 , «_2) - X. = 0, I 



D2(M_2, 1^-2) + Y = 0,1 S2(«-2.'^-2)-Y. = 0, f • ■ . . (1^; 



and 



DsO^-a) + Z = 0, 83(w_2)-Z^ = (13) 



Now (12) are simply the equations of plane strain under body forces X, Y and 

 surface forces X^, Y^. They have no solution, except a rigid body displacement, unless, 

 in the notation of Art. 38 (1), (2), 



2X = 0, 2Y = 0, 2(a;Y-2/X) = 0. 

 Also (13) have no solution except w_2 = constant, unless 



2Z = 0. . 



(14) 



(15) 



In the general case, the conditions (14) and (15) are not fulfilled. Hence the 

 applied forces cannot be brought in at this point, or p is greater than — 4. Conse- 

 quently w_2, v_2, w_2, are given by 



[W_2, V_2, W.i\. 



Di(m_2 , W_2) = 0, ) 

 = 0. ) 



Si(m_2 , ^-2) = 0, 

 D2(m_2 ' ^-2) = 0, ' 8^{U-.2 . ^-2) = 0> 



(16) 



D3(«'-2)=0. 83("'-2) = 0- 



