EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 917 



As in Art. 5, we subtract these from (7) and then integrate as to k from to co 

 We thus obtain, say, 



^"' = - — I { {Gmixp - A^ m-JmiKpyTmiKp COS k{z - z) - -— ( i- - <^„ > -^^^K COS m{lo - co'), ^ 

 '' TT Jo ( Am\p/ ) K^ 



"'2= - — I 1 - B^, „iJ„i'iKpJ,„2.'/<p' COS k(2 — a') - ^„ > —^/k cos ?;t(a) - (!>'), I. 



l/',^'' = - — I ] - C,(,, m-^ miKp-l iniKp' COS k:(z - z') - i/',, V —(7k siu ??l(a) - w'), 



p>p. 



The strain defined by (9) and the slightly altered form for p<ip' has the singularity 

 (1), but is otherwise regular within p^a, and it yields null stress at jo = a. 

 By the transformation 6 (7), (9) becomes 



,m a {[^ Bp A^ mj /3p\. Bp -«^i:£lU ,„ / '\\ ^ 



in 



Hence, as at 6 (9), 



,m 



^ "J„,'^)J,„^^e a —//^ COS TO(a) - <o'), 



Dm a I a p- 



% '"J„;-^)j-^e-'-^^lrZ^ sin m{^ - <.'), 

 Dm a J a p^ 



third 



and 



fourth 



paths. 





"f l3HD,nld(i\ C^, 



1 2ft 

 + coefficient of „ in 



a a sm 



- p ^p A^ „i j Pp y>,i,,i,tT I3p 



a Dot 'I I* 



C,(,, m T ^P 



-:;- tljn 





/8 /^ 



m a 



In (11) p'>p' and £>/ ; for p<ip' or 2 <2' interchange p, p or z, 2' 



In the same way we find a solution for the source 



9=1 ( - - ) < cos k(2 - z'),i„{iKp'GcjKp - ^— ( - ) \ -'Ik cos m{<x) - w') 

 jo V TT/ ( 2m\p/ J K- 



in the form 



cos 

 sin 



, Hi /jW 



-A 



V ^^^ J^«' "^' ^«. ™ 'V,„^J„,^'e"^^ ""' m(co - 0.') 



'jihin,nld[i\ Ce, 



a a 



+ coefficient of -- in ^ 



/Ae,OTT /3p ^ f3p ho. Ill T /^p N 

 ^ ;„ ^'*/ D,,, a a D,,. ft 



Ce,TOT pp 



^pr — Om — 

 D,„, ft 



a sin ^ ^' 



and a solution for the source 



in the form 



■ m /im 



mz-2'). 



/ jLjB'hiDm(l^\ C-A/'" J a a cos 



\ a Dm a, ' 



(10) 



(11) 



(12) 



(13) 



(14) 



(15) 



