EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 977 



Consider now the series 



2^ I ' '" ^P cos k{x - z')iiK {p<a). . . . . . . (17) 



m Jo JjJKU 



The modulus of the general term does not exceed I of (4). 

 Hence the series (17) converges at least as rapidly as the series 



2v/m(^)"'. 

 m \a / 



Thus (17) is absolutely and uniformly convergent within a cylinder of radius «-»?, 

 where >y is fixed, but as small as we please. With the help of (12) and (13) the same 

 may be proved of the series of term-wise derivatives of (17) of any order, with respect 

 to the coordinates. 



Convergence of the series in the elastic problems. 



The series in the elastic problems involve more complicated general terms than that 

 of (17) in the preceding Article, but they may be treated by a similar method. 

 Take as an example the solution for a unit radial force at the surface (Art. 6). 

 Here 



<^m = ?- I i -^r-'SJxp cos k{z -z')-fL- -— — \dK ■ COS m ((O - w') . . . ( I ) 



= _— j i —Lj±k^J iKo COS k(z -z) -^— — — \d\ - — ) • cos m((ii - co') 

 T>Jo I D,„ '^ a'" 2a- J \ k/ 



= _— - / — ~^\ ^■"' k^J,Jkp cos k{z — z) \dK. ■ cos m{iiy — w') 

 tt'-llJo K dK\ D„, } 



= 4-(Ii + I2) cos ?h(o) - w'), (2) 



where 



T, = I —{-^ k'^J,Jkp ) cos k{z - z')dK, 



Jo K dKXT) J 



T / ,v /'°°sin k(z - 2') A, 9T . , 



Jo K D 



(3) 



In (3) write ima for iKa{ =/3). 



Then, using Art. 50 (2) in Art. 30 (l). we find 





Thus, putting a = at so that k = mt, we have 

 Then, from (3), 



K^AiT • • 1 SJ.mpf ,,, 



— -J J^t/cp = — 5 *"■ ^ (5) 



D 2a^ J,„imat 



L = - (z - 2')— r Ji«! sin {m(z-z')t} ^, 



(6) 



