EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 937 



7j force at {p , «', z'). 

 Take Po = 0, Oo = 0, Zo-1. 



for 5;>2' ; for z<z\ interchange z and 2', and change the signs of 0, 0, and >//. 



29. The transitory modes for an internal force in terms of displacements. 

 Write the displacements corresponding to 



in the form 



<^i,™ / sin ^ ^ ""a 



■"»:")K''"t?;)'"''>"'^'">'<"-«''- ■ 



Then in the preceding Article Up{p') : Up(a) = Up,i{/) : Up,](a). 



Use a similar notation for the solutions proportional to this, but involving A, 

 B, and C with suffixes 2 and 3 instead of 1. 



Thus the results of last Article become for the general transitory mode : — 



'P force at {p, co', z'). z^z'. 



(»-■''.)= __^._„^(U...W.yM(p)A-fci>co3 



\ u^ J TTfjidBJdfi Up,i(a)\ Lla,,i(/D) / sin ^ ' ^ ' 



Deforce at {p\ w', z'). z'>z\ 



('";:'•)' - m"„« #4^;r''*>- VvW').-^'(--'") .,(.-.;). . . (2) 



\ «« / TrfjLdD.Jd/3 U„,2(a)\ L'„,2(p) / cos ^ ^ ^ ^ 



Z force at [p , co', z'). z^z'. 



\ n-> ) TTfj.pdDJdli Uj, 3(ffl)\ Ll„,3(|o) / sin ^ ' ^ ' 



By 27 (13) it can easily be shown that (2) and (3) may also be written in the 

 forms 



TTfj^dDJd/S Up,i(a)V U<,,i(p) ; cos 



u,, u,,\_ , Zc„, Ue,i(p')^Up,i(p), U,,i(p),\ -«^„--^)cos 



W„: 



7rixdDJd/3 Up,i(a)\ Ua,,i(/3) / sm ^ /> v / 



which are more convenient for comparison with (1). 



The forms (1), (2), and (3) allow an immediate verification of two important 

 properties of the solutions, viz. : — 



(i) Wp at (/, ft)', z') due to a unit P force at {p, ft), z) = u^ at (/>, ft), z) due to a unit 



P force at {p, ft)', z') ; and similarly with u^ and u,. (Cf Art. 24.) 

 (ii) For p' = a, the solutions reduce to the solutions for surface traction, given 

 in 6 (11), 7 (3), 8 (3). 

 In addition to the relations (i), three other relations follow from Art. 24 ; these 

 are easily verified from the forms (1), (2)', and (3)'. 



