EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 957 



The form of the condition at (9), (10) shows that if F{z) be fixed while a tends to 

 zero, the term -by -term integral of (5) with respect to z' has the better chance to 

 converge the smaller a is. But of course it may diverge no matter how small a is. 

 It is then an asymptotic series. The sum of p terms diflfers from the value of u^^ ,„ in an 

 exact particular solution by a certain remainder. This remainder does not vanish for 

 p infinite, but it involves the factor a^^~'\ so that its order in a continually increases 

 with p. 



In our solution the remainder is given by a definite integral, derived from the 

 last line of 39 (4) by integration as to the accented variables over the section of the 

 cylinder. 



41. Nature of the error in the approximate values of displacements and stresses 

 defined in Art. 38. 



The method of the two preceding articles may be applied to force and displacement 

 in any direction. When force and displacement are both perpendicular or both 

 parallel to the axis, the series corresponding to 40 (5) will involve only even z- 

 derivatives of P(2) ; otherwise it will involve derivatives of odd order only, for 

 the integrals of 39 (l) will then be connected by the minus sign. 



The complete solution for the whole cylinder may be considered to be composed 

 of two parts, viz. : — 



(i) The asymptotic particular solution, the leading terms of which are defined 

 in the tables of Art. 38. Outside of the region of applied force, this part 

 of the solution contains only a finite number of terms, multiples of 

 permanent free modes. The order in a increases by two from term to 

 term of the series for displacements and stresses. 



(ii) The transitory perturbations (end of Art. 39), the influence of which is 

 insensible except at points whose distance from the sections z^ and z^ where 

 the force, or some of its ^-derivatives, is discontinuous, is of the order a. 



The office of these perturbations is obviously to secure the continuity of 

 displacement and stress at the critical sections. 



The order of the displacements and stresses at these sections due to the perturbations 

 is easily made out from 39 (4). There the leading terms in the second and third lines 

 are of the same order as the leading term in the first line. But when force and 

 displacement are one of them parallel and the other perpendicular to the axis, the 

 second and third lines will be of lower order by one than the first. 



(a) It easily follows that the displacements due to the transitory perturbations 

 are of higher order by one than the displacements of highest order in the tables of 

 Art. 38 ; that is to say, they are of order a^. 



(h) With respect to the order of the first derivatives of the displacements, there 

 is an important distinction between the transitory part (li) and the main part (i) of the 



