EQUILTBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 949 



Substitute the value of m^ from (4) and (5). 



The integration in (7) can be taken term by term [cf. Art. 33), so that we obtain, 



for z^<z<z^, 



[z { \ 1 U ^^-^') I [^ {\ 1 U &i^ I 



1=1 < - coefficient of - in -^e " r Pf^J^' +1 ■( t: coefficient of - in — ^g a \ Fdz' 



U C fz P(Z-Z' ) ^ rz.2 «2_-2') 



+ 



for z>Zo, 



dBjdfS 



( fz P{Z-Z1 /•2., «Z-2') ) 



< jPe a dz'+ "Pea dz ^ 



(8) 



and for ^ < Zj , 



I^/Jlcoefficientof^inU..^}p.z',-Z{^-^/;p.~^.z'[ . 



(9) 



Now the first or permanent part of the value of u^ in (4) is just the u^ of the 

 permanent source terms collected in Art. 19. It is of course zero if m>l. 



Thus when we substitute the value of I from (8) or (9) in (6) the part of the 

 displacement in (6) arising from the " coefficient of 1/^" terms is just what comes from 

 integration of the permanent terms of the point source solution ; and similarly for a 

 force or displacement in any other direction. In next article we shall consider 

 this part of the solution for any distribution of force. 



The part of the solution arising from the 2 of (8) or (9), that is, from the transitory 



terms of the source solutions, will be considered in Art. 39. 



38. Solution for any distribution of body or surface force — the part arising from 

 the permanent terms of the source solutions. 



For this part of the solution we shall suppose the applied force expressed in terms of 

 X, Y, Z instead of P, O, Z components, on account of the greater symmetry of the 

 results. And for the same reason the components of the displacement are taken 

 parallel to the rectangular axes of x, y, and z. 



The results for the (point or surface) source solutions are collected in Art. 25 for 

 2;>2' ; for z<z' we have to change all the signs. 



The body force per unit volume is X, Y, Z ; the surface traction per unit area 

 X^, Y^, Z^. The only difference in the results for the two varieties of applied force is that 

 one appears in volume integrals, the other in surface integrals ; and for the sake of com- 

 pactness we represent the sum of an integral over a section and a similar integral round 

 the contour of the section, of the type 



jf{a^',I/',z')X(x',y', z')dx'dy' + jf{x, y',z')X^{x',y',z)ds', . . . ■ (1) 

 by the symbol 



'^{f{x',y',z')Xix', y',z')} simply (2) 



The results are best expressed in terms of certain functions of z only and their 



