EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 907 



Thus (1) is equivalent to (11) within the cylinder if 2>2;'; the corresponding 

 equivalent to (l) when z<2' can be seen at once from (4), (5), (6), (7) to be obtainable 



from (11) by changmg e" « into e a . 



The two forms for (p, 9, or ^ in the regions 2>2' and 2<2' respectively represent 

 one and the same function (l), which is regular for all finite values of z within the 

 cylinder. 



Fo7- a unit element of 7'adial traction at (a, w', z') 



(<^, 6, v)=(<^p,,^p.,V'iv)=2:'(<^;;',,6^;;,^pj (12) 



7. Transverse traction. Solution in series. 



Take - — of the expressions in (1), (2), (3), Art. -5, and it is seen that 



711, 9 ft) 



6, 9,\ a- 1 /A., „. , B., ,„,\ r . , ,. (-sin) / ,>. y-,\ 



^' , ' =— -"'I - ]j ,,,'lKp con k(Z - X ) ^ 'Vtiw-ID) . . . (I) 



•A / //. D„A ^2.,n J '" ' ^ ^ COS ^ ' ^ ' 



give, at p = a, 



pp = 0, pw = cos Vl((x) - Oj') COS k(2 - z), />Z = 0. 



Comparing (l) here with 5 (1) and 5 (5), we can write down at once the result 

 corresponding to 6 (ll), (12), viz. that /or a unit element of transverse traction at 

 («, ft.', z') 



{<l>,0,f) = ^i<t^Z,Ol,^'^^), (2) 



where 



/•^l.flA _e,y . 1 (A, E,., I ^Jp^-^M(-sm) , 



^ . coefficient 01 — in ^ — < '-'A io„P^e » ' j/<(aj - oj ), 



27r/x /3 D,„l C,.„, j '"a cos ^ ^' 



for 2>z' ; for 2<2' the same with 2^ — 2 instead of 2 — z! . 



(3) 



1 7s 



Take - — of the expressions in (1), (2), (3), Art. 5, and it is seen that 



8. Longitudinal tractio7i. Solution in se7'ies. 

 1 d_ 

 K dz 



give, at jo = a, 



pp = 0, fHo = 0, 'pz = cos m((u - (o') cos k(2 - z). 



Following the lines of Arts. 5 and 6, we find for the m-component of a unit 

 element of Z traction at {a, w , z') 



( '^^' 5' ) = -^ /-I- ( ^^■™' ^'■™' i J^^e-^^'a '^^f' m(a> - <^')dp, (third and fourth paths). (2) 

 The form (2) is convenient for changing to the series of residues at and to the 



