EQUILIBRIUM OF AN ISOTEOPIC ELASTIC ROD OF CIRCULAR SECTION. 943 



With the help of the asymptotic forms we can investigate the behaviour of the 

 various series of terms of the transitory type for displacements and stresses with respect 

 to convergence, when z is put equal to z' in every term. 



In Art. 31 (6), (7) the series converge for all the displacements, and all the stresses 

 except pj. 



In Art. 32 (5) the series converge for all the displacements, and for the stress jow. 



For internal force, 31 (9) and 32 (7) show that comparatively few of the series 

 converge. 



Hence, in order to derive solutions for extended distributions of body or surface force, 

 we have to proceed by the following method, or some modification of it, in the case 

 when the section z lies within the region of applied force, i.e. when 0i<3<Z2, say. 



Imagine a slice of small thickness between the sections z^ and Zj^ cut out, that is to 

 say, let the applied force vanish when z^<z<Z/^. 



Calculate displacements and stresses at z, and integrate the resulting series, 

 multiplied by the applied force, term by term. Then take the limits for ^g and z^ 

 tending to z. 



It can be shown that the limits can be taken term by term, and that the resulting 

 series for displacements and stresses are the exact solutions of the problem, provided 

 the applied force is finite and integrable, and for z fixed is a function of /> with 

 "limited variation." * 



Note that we get a different form of result for the stresses if we first take the limits, 

 for Zg and z^ tending to z, term by term and then differentiate term by term as may be 

 necessary in order to calculate the stresses. The two forms of result can be shown to 

 be equivalent. 



34. Order of magnitude of the displacements and stresses in the source solutions, 

 when the radius of the cylinder is considered to be an infinitesimal of the first order. 

 Distinction between the permanent and the transitory terms. 



In Art. 16, the auxiliary source functions (p, 0, \/^ have the form of a product of a 

 by a function of x/a, y/a, z/a and x'/a, f/a, z'/a. Then the formulae of Art. 19 give 

 (p, 0, ^ for a unit force in any direction as the product of l/fj- by a function of these 

 variables. 



Lastly, the formulae of Art. 2 (7), (8) give each displacement and stress as 



a product of 1/m«, -^ respectively by a function of x/a, y/a, z/a and x'/a, y'/a, z'/a. 



That any displacement due to any force of F units must have the form 



F //x xj z_ .t' y z 



fji.a \a a a a a a , 



may be inferred from the consideration of physical dimensions ; for ¥/fj.a is a length. 

 More rigorously, if we write (^, 'i, ^ ; ^', *]', '(') = {x/a, y/a, z/a ; x'/a, y'/a, z'/a), and 



* PiERPONT, The Theory of Functions of Real Variables, vol. i. p. 349. 

 TRANS. ROY. SCO. EDIN., VOL. XLIX. PART IV. (NO. 17). 129 



