EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 897 



complete values of the displacements is next in importance to that from the straining 

 permanent modes. 



An independent method for calculating the coefficients of the permanent and 

 transitory modes in the solutions for a single force is given, based upon an application 

 of Betti's Theorem, of which theorem much use is made throughout the paper. The 

 chief value of this method is that by its means the permanent terms of the solution for 

 any force can be found for any cylindrical boundary for which St-Venant's solutions 

 are known, if only we assume, what can hardly be doubted, that any source solution 

 can be expressed in terms of permanent and transitory modes analogous to those 

 for the circular cylinder. This apjjears to be a more satisfactory method than any at 

 present in vogue for determining the approximate character of the deformation of any 

 beam at a distance from the ends. 



Solutions for any distribution of force in the body or at the cylindrical surface are 

 deduced from the source solutions by integration with respect to the coordinates of the 

 source. With the help of successive integrations by parts with respect to the variable 

 z', the solution is exhibited in the form of series proceeding by terms whose order in 

 the radius increases by two from term to term. These series can be stopped at any 

 point, an expression for the remainder being given. If continued to infinity they 

 may be convergent or only asymptotic, and a sufficient condition for convergence is 

 investigated. 



It is shown that the successive terms of the asymptotic solution can be deduced by 

 a purely algebraic process from the complete solution in its original form before the 

 transformation by Cauchy's Theorem is applied. 



At any section, say a critical section, where the applied force or any of its 

 successive z- derivatives is discontinuous, perturbations of the transitory type make 

 their appearance in the course of the process of integration by parts. These perturba- 

 tions secure the continuity of the displacement and stress at the critical section. 

 The order in a of the displacement and stress due to these perturbations is easily 

 made out. 



A table is given of the leading terms in the displacements and stresses in the 

 general solution for an infinite cylinder. On the assumption that the body force per 

 unit volume is of order zero in a, and the tractions per unit of the cylindrical surface of 

 order one, this table gives explicitly all the displacements up to order one, and all 

 the stresses up to order zero, inclusively. 



It is shown with the help of Betti's Theorem that any deformation of a finite 

 cylinder due to force at the ends only is expressible in terms of the permanent 

 and transitory modes of the infinite cylinder, the influence of the transitory modes 

 being confined to the neighbourbood of the ends ; and of course the solution for 

 the infinite cylinder gives a particular solution for force in the body or at the 

 cylindrical surface. 



An approximate solution of any problem is thus obtained involving undetermined 



