This paper presents a general analysis of the dynamic stability of a finite-length, fluid-conveying pipe. The Flu¨gge-Kempner equation is used in conjunction with classical potential theory so that circumferential modes as well as the usual beam modes may be considered. The cylinders are found to become unstable statically at first but flutter is predicted for higher velocities. The critical flow velocities for short, thin shells are associated with a number of circumferential waves. This number reduces for thicker and longer shells until the instability is in a beam mode. When the limiting case of a long thin shell is taken, it is found to agree with previous results obtained using a simpler beam approach.