EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 975 



50. On the convergence of the series defining the exact solutions (Arts. 6, 7, 8, 19) 

 J'o7' isolated forces. Preliminary. 



Bach series is of the form 2'^^, and the nature of its convergence may be investi- 

 gated by considering the asymptotic form of t^ for m infinite. (If we can find a 

 function T^ such that the ratio of T,„ to t,,^ tends to unity as m tends to infinity, where 

 T^ has a simpler analytical form than t,,^, we say that t,„, = T,^ asymptotically, or that 

 t ^T ^ 



'^m — ■'-111' J 



The investigation may be based on the asymptotic forms of J„,(^mo), G^(»wa), where 

 m, a are real and a is fixed. 

 It may be proved * that 



J„Xtma) = 45^(«' + l)~*e'"'^ { 1 + .4- A^ + 



sjiirm 



24m(a2 + l)« 



G„j(ma) = TT 



l-"'a-\ . ,x-..-,„A \ . 1 3a2-2 



where 



v/2 



27rm 



:(a2+l)-ie-'»'^^ 1_ 



I 24m(a2+i; 





A = Va2+l-log(l + Va2+l). 



(If we write J = —j=f{m) , then G = — ^/( — w). ) 

 From these 



(1) 



JJ{ima) _^ Jo^ + 1 1 ia 

 J,„(?'v«a) ia 2m a^ + 1 



G.„'(i7na) _^ Ja^ -1-1 1 ia 



2m o? + 1 



(2) 



(j„,{ima) 



The latter results may easily be verified and extended by means of the differential 

 equation for J and G. Further, if we write /3 for ma in (l), we obtain asymptotic series 

 for Jm(*73), GrJi^) valid not merely when m and ^ are large in a finite ratio, but 

 when m? + jS^ is large, whatever the ratio of m to /3 may be. 



If we write ■m = R cos 0, wa = R sin Q, we get 



JKcosa(*R sin ^) = --^(j-J^jV^e«{l +2^(3 sin2e-2cos2^)+^ + ^^ 



where W2(^)> "^sl^)} • • • ar© rational integral functions of sin and cos 0. 

 We may now find the asymptotic form of 



}, (3) 



l=ri^dK (p<a). 

 jo J,jKa 



(4) 



With mt for k, this becomes 

 But, from ( 1 ), 



l = mf^Jr^dt (5) 



ja J,„imat 



J,Jmaf \aj \l+pH^J 



e-"'T(l + e), 



(6) 



* By a similar process to that given for J,„(ma;), where x is real, by Graf und Gubler, Einleitung in die Theorie 

 der Bessel'schen Fimctionen, erster Heft, Bern, 1898, S. 96. 



TRANS. ROY. SCO. EDIN., VOL. XLIX. PART IV. (NO. 17). 133 



