EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 909 



Longitudinal traction. 



1 ^ 1 , «p _«5z^) 



('^' ^) = - 9^?;^n:^(2/3J", 2 - v^J - 2^2J')J„^^e « 



(3) 



1 1 1 /Jo _«iz^) 



-^ — . coefficient of - in (2/3J", 2 - v/3J - 2j82J')J,^e « , 



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

 The summation extends over the zeroes of Cjo with positive real part. 



10. Jlie permanent terms in the solutions for surface traction, unit element. 

 These are found by ordinary algebraical expansion as indicated in the various 



formulae 6(11), 7 (3), 8 (3), 9 (1), (2), and (3). 

 We recall the expansion of the function J, viz. 



T °''" i 1 "^ °^^ "" i /'1\ 



'«* = 2^m I " 2(2m + 2) "^ 2^(2m + 2)(2ni + 4) ~ 2.4.6(2m + 2)(2m + 4)(2?» + 6) "^ • • • j- • { ) 



As some of the leading terms in the expansions of D,„ and its first minors vanish 

 when TO = or 1, it is best to consider these cases separately. 



In all the cases it will be seen that the permanent parts of the values of (p, 0, and "^ 

 in the two regions z>2:' and z<2' are the same functions but with opposite signs. 



We need therefore only write down the forms for 2>2'. 



11. Permanent term,s for unit element 0/ surface traction, m'^1. z^z' . 



D,„ = 4m2(m-l)/3'"'+7(2™nm)3.* . . . ' . . . (1) 



'A,,,„- -2m2(TO-l)/i='", A,,,„= -2?n2(TO-l)y3-"', A3, „, = 2m(w - l)/3-™+^ \ ^^]i^ 



Bi, ,„ = - 2hi2(to _ 1 )(,tt + v)/i-'«, B.,, ,„ = - 2m2(m - 1 )(v// + v)/3'-"', B,,, ,„ = 1m{m - 1 ){m - 2 + v}P^'"+\ l(2"'nm)2. 

 IC,,,„ = 2m2(«i-l)(m + v)/32"', C,.,„ = 2?tt2(TO - l)(m + v)/?-'", 0^,,,,^ -2m{m-l){m-2 + v)fr''"+''j (2) 



Normal traction, 6 (H). 



1 / ,^ „, cos , ,.f\,m + v\ ,„, 



"A / 47r//.ffl'"+''" ■'"' Sin '^ ' ''\-m-v_ 



Transverse traction, 7 (3). 



\^ ) 47r/x«'"+^^ '^ COS ^ \-m-v) ^^' 



Longitudinal traction, 8 (3). 



f4>,0,\ 1 ,„ cos , ,,/l, m- 2 + 7'A ,^. 



\ ^ J iTTfima'"' sin ^ \-m + 2-vJ ^^ 



If these values be substituted in the formulae 2 (7), it will be seen at once that they 

 give no displacement. 



* In this and next Article, (1) and (2) state the terms required in the expansions of the various functions in 

 ascending powers of 0. 



