256 



Trans. Acad. Sci. of St. Louis. 



y = 



x^J^ ( u' ) 



nj.in,') —I {—ay J Jin;) 



+ x 



.(30) 



2 3. i 2 3. i 



in which w' = ^x^{ — a)^ and m/ =3?^( — ^y • 



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

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

 X = Z in (29) and y = J/ for ic = Z in (30), we have 



PJ,{u,) 



A, — 



I 2Jj^ (wj — a^lJ^(u^) 



(31) 



?V,(w',) 



+ ? l (32) 



r^jju;) — i(-ayji iu\) 



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

 a; 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: — 



3, ./ 1 



A ~ 2 ^ tan^ 



1 — Jpx 



-) 



(33) 



in which 



X= ' 



3 



Now by the calculus of Gamma Functions 



r(p+ i)=pr{p), 



