978 THE EQUILIBRIUIil OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 

 Hence 



where 



Also 

 where 



Iil<l3> (7) 



^ 2ahn \aj Jo i \\^pH'^; 



V^ 



2-p2) 1 /pV" 



m. 4a- V a J ' 



\h\<\, ■ -• • • (9) 



2a2 



From (2), (3), and (7), . . • (10) the absolute and uniform convergence of the 

 series 2'd),„ is obvious. 



With respect to I2 we may get a more definite result. For if we differentiate the 

 second integral in (6) and note that -^. — — is a decreasing function of t (as may easily 



be verified from (6), (9) of last Article), we see that that second integral in (6) 

 increases from 2=— 00 toz=cxD. 

 But its limit for 2 = co is 



2 \3 ,nimat ) t=a 



Hence, as 2 goes from — 00 to^oo , 



-^ goes from -t^„( -^ ) to - ■^I—j ■ 

 z-z 4a2\ a J 4a-\ a / 



It is easy to see that Ii vanishes for 2 = dzoo . 

 Hence 



di„, = + ( — ) (2 - !2') COS m(o) — to') 



for 2 = ±:cx) . This agrees with Art. 11 (3). 



