EQUILIBEIUM OF AN ISOTEOPIC ELASTIC ROD OF CIECULAR SECTION. 955 



When a is small, the corresponding deformation is insensible except very close 

 to the section z = z-^. We may call it a transitory perturbation. 



Similarly with the third line of (4) at the end 2 = 22- 



We will return to these transitory perturbation terms in Art. 41. 



Lastly, tho, fourth line of (4) contains a series of the same general character as the 

 series (1), but it is at once of higher order in a on account of the factor a", and more 

 highly convergent on account of the factor /3"" in the general term. 



40. The asymptotic particular solution for body and surface force. 



The complete value of I in 37 (8) consists of the first line there and the four lines 

 of 39 (4). We consider in the present article the first line of 37 (8) in combination 

 with the first line of 39 (4) ; let the sum be denoted by S„. 



By 39 (7) 



Also 



Dm ~ VI ' m" m " 



-'-'4;^_i ^(^-^') I i^Kz ~z'f _ i^\z-z'} 



2a2 



6a3 



and 



p{z-z') 



F^ = 1 + ^(^^) + ftnlIV &z£T 



+ . . 



(1) 



(2) 



(3) 



(4) 



2a? 6a3 



Hence the first line of 37 (8) is equal to 



+ r { A'*"UL-^'(^-^')^+ \a-^^\:^\z-z') I V{z')dz', 

 or, in the notation of 38 (6), to 



These terms do not occur if m>l. 

 Hence 

 S„ = (4)liere + 39(8) 



= _ a-3U<-^'D-*P(z) - a-iU'-2'D-=P(z) - aU™P(2) - a^U'^'-^Pfz) - .... to the term in a"-i or d!\ ( 



Denote the last line of 39 (4) by E,„. 



As we saw at 39 (9), the second and third lines of 39 (4) correspond to free 

 transitory modes, which contribute nothing to body or surface force. 

 Thus, by 37 (6) and (7), if we write 



(6) 



I 



(5) 



\ fa C^TT 



U = \ \ (Sn + Rn)p'(^jo'(ia)', 



p'"* yo Jo 



then u^ ^^ is the jO- displacement in di particular solution for the body force and surface 

 traction, for z-i<z<z^. 



The functions U(;^\ U(^'>, U^\ W^, etc., of p, «, /, w' are all of order a'^ (Art. 37 

 after equation (3)). Hence in S„, or in its term-by-term integral over the section z, 



