Paidoussis 

 - -|[ ML(x,f, - xgfg) + mL(x, - x^)] Vc " \ MUL(f , + f2)yc 



+ Mu'l [\ ^c,p - ^^2*^] cj> = 0, (16) 



where Cjp= CyL/D + c, + Cg. The sarae equations could have been 

 obtained from first principles. It is noted that here L is the length 

 of the cylindrical portion of the body which is smaller than the over- 

 all length, as used in §5, by i| +^2* ^^^ difference never exceeding 

 a few per cent. 



We non-dimensionalize these equations by introducing t) = yj./L, 

 A = s/L, e = L/D, X| = ^i/L» Xg ~ ^9/^ ^'^^ ''' ~ Ut/L, and consider 

 solutions of the form 



itJT _ iwT 



r) = He and (j> = ?>e 



where co is a dimensionless frequency defined as to = Qih/U, S2 

 being the complex circular frequency of oscillation. Substituting r\ 

 and <j> into the non-dimensionalized equations, and noting that our 

 assumptions require m = M, we obtain 



j[2 +Xi(l+f|)+X2(l+f2)](-'^') +[-^ec^+f, -f^lM) +[-|c^/A][h 



+ j--|[x,(i +f,) - X^i^+i^m-^^) +[2--|(f,+f2)]M 



+ [-| ec^ + f, - i^-{ (1/A)c^pl [^ = 0, (17) 



and 



j - [ "I X, (1 +f,) - 4 Xad +f2)](-<^) -[ \ (f, +f2)] M) - [ \ c,p/A] ( H 



+ [i(i/A)c^p-l (f, +f2)]|^ = 0, (18) 



1004 



