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XIX. On the Instability of Cylindrical Fluid Surfaces, 

 By Lord Rayleigh, Sec. R.S* 



IN former papersf I have investigated the character of the 

 equilibrium of a cylindrical fluid column under the 

 action of capillary force. If the column become varicose 

 with wave-length X, the equilibrium is unstable, provided 

 A, exceed the circumference (27ra) of the cylinder ; and the 

 degree of instability, as indicated by the value of q in the 

 exponential e qt to which the motion is proportional, depends 

 upon the value of X, reaching a maximum when A=4\51 x 2a. 

 In these investigations the external pressure is supposed to be 

 constant ; and this is tantamount to neglecting the inertia of 

 the surrounding fluid. 



When a column of liquid is surrounded by air, the neglect 

 of the inertia of the latter will be of small importance; but 

 there are cases where the situation is reversed, and where it 

 is the inertia of the fluid outside rather than of the fluid 

 inside the cylinder which is important. The phenomenon of 

 the disruption of a jet of air delivered under water, easily 

 illustrated by instantaneous photography, suggests the con- 

 sideration of the case where the inside inertia may be neglected; 

 and to this the present paper is specially directed. For the 

 sake of comparison the results of the former problem are also 

 exhibited. 



Since the fluid is supposed to be inviscid, there is a velocity- 

 potential, proportional to e ikz as well as to e qt , and satisfying 

 the usual equation 



pk + lf- k ^=o (i) 



air r dr 



If the fluid under consideration is inside the cylinder, the 

 appropriate solution of (1) is 



4>=J (ikr)=Io(kr); (2) 



and the final result for q 2 is 



2 _ T (l-tfa*)ika.J '{ika ) 



~~ pa 6 J (ika) 



_ _T (1-Pa 2 )kal '(ka) 



~pa 3 I (ka) ' ••■•<«>; 



* Communicated by the Author. 



f (1) " On the Instability of Jets," Math. Soc. Proc. November 1878. 



(2) " On the Capillary Phenomena of Jets," Proc. Roy. Soc. Mav 



1879. 



(3) " On the Instability of a Cylinder of Viscous Liquid under 



Capillary Force," supra, p. 145. 

 Phil. Mag. S. 5. Vol. 34. No. 207. Aug. 1892. N 



