256 



Trans. Acad. Sci. of St. Louis. 



y 



a 



xVi(w') 



I 'JJO -H-ayjjn;) 



+ x 



.(30) 



2 3. i 2 3 



in which w' = oa3^( — a)^ and w/ =-ol { — a) 



i 



Now if we put Jj and J/ for the deflections of the upper 

 and lower halves of the tube respectively, since y = Jj for 

 X = Z in (29) and y = J/ for x = I'm (30), we have 



^. = — 



PJAu,) 



I ^J^{u^) — a^lJ^(u^) 



(31) 



1 n % 



PJju\) 



a 



H, 



J^{u;) — i(—ayji (u\) 



+ I \ (32) 



By evaluating the Bessel's Functions the values of J^ and 

 J/ may be computed from (31) and (32). More convenient 

 formulae may be deduced as follows. Substituting for a and 

 b their values as given above, Eq. (31) may be written in 

 this form: — 



in which 



J^ = 2 ? tan^ 



{ 



1 3 



— 1 



\1 — a"P -X 



X = 



3 



(33) 



Now by the calculus of Gamma Functions 



r(p+ 1) =pr(p), 



