256 



Trans. Acad. Sci. of St. Louis. 



x 2 J 1 (u') 



I 2 J JL (u 1 ') — l(— a) 2 J±(u;) 



3 3 



+ X 



(30) 



2 3 _^ 2 i 



in which w' = o^( — a ) 2 an d w/ = qJ ( — a ) 



Now if we put A 1 and J/ for the deflections of the upper 

 and lower halves of the tube respectively, since y = A 1 for 

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



b 



A, = — 



PJAu x ) 



1 I 3 J x {u 1 ) — a 2 U ± (u 1 ) 

 3 3 



(31) 



l 2 J x (u\) 



a: = 



rt 



,/•(„/)-_ j (_a)Vi («',) 



+ I \ (32) 



By evaluating the Bessel's Functions the values of A 1 and 

 &l 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: — 



4 = £ I tant 



in which 



X = 



1 — a 2 l 2 X 



JA u i) 



3 



Now by the calculus of Gamma Functions 



(33) 



r( P + i)=pr( P ), 



