EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAE SECTION. 947 



(ii) the applied force has the form ¥{x/a, y/a, z), and, in particular, the extreme 

 values of z, beyond which F vanishes, are independent of a. 



We will now consider one or two cases where these conditions are not fulfilled. 



(a) If the applied force is the sum of a finite number of functions satisfying (ii), 

 we can arrange these functions in a series of terms of ascending order in a, and deal 

 with each separately. This method may be applied also when the ascending series is 

 infinite, or asymptotic. 



(6) Let the extreme values z^ and 2.2 differ by a multiple of a, so that 2;, = Zj + joa, 

 where p is independent of a. Thus in effect the condition (i) fails. 



Put z = Zi + a^, z' = Zi + wC^ ; thus ^ and X^ are of order zero. 



Then 



r(izilF(2>2'=aPa-rm.,+^^'Mr, (i) 



Jtx aP Jo 



while 



|3(Z-Z') 



F{z')dz' = a 



Jo 



I e « Y{z)dz' = a\ e-^^-^"^(z^ + aOd^ (2) 



Ai Jo 



(1) and (2) are in general of the same order, so that the contributions from permanent 

 and transitory source terms to any displacement or stress are of the same order at points 

 within the region of applied force. Beyond the region of applied force, however, the 

 contributions from the permanent source terms still predominate. 



(c) Let 01 and Zg be independent of a, but let F have the form Y{xla, y/a, z/a). 

 To fix ideas, suppose F to be a periodic function with period proportional to a, say 



inz 



F = 2A„e^. 



rz in£ rZ _p(z-z') in£ 



Now j e "'dz' and / e "~^^e^dz' are of the same order, so that in this case the con- 



tributions to the solution from permanent terms independent of z in the source solution 

 are of the same order as the contributions from the transitory terms. 



Hence only those integrated permanent terms are reliable which to begin with 

 contain a power of 2 — z'. In particular, the only stress obtainable to a first approxima- 

 tion from the permanent terms of the source solutions is the stress zz, and that in the 

 case of flexure (m = 1) only. 



37. Deduction of solutions for a body or surface distribution of force from the 

 point source solutions. 



In Art. 16 (10) the solution for the m-component of a source ^ = J);\~^ is expressed 

 in terms of integrals over the " third and fourth paths " (Art. 6), and similar forms 

 can be written down for a 6 and for a -^ source. 



Let displacements be calculated from these "third and fourth path " integrals for 

 any given unit force by the formulae of 2 (7), and 15 (17) modified as explained in 

 Art. 18. 



