252-253] INSTABILITY OF A JET. 461 



swellings and contractions of this wave-length, with continually 

 increasing amplitude, until finally the jet breaks up into detached 

 drops*. 



253. This leads naturally to the discussion of the small 

 oscillations of a drop of liquid about the spherical form-(-. We 

 will slightly generalize the question by supposing that we have a 

 sphere of liquid, of density p, surrounded by an infinite mass 

 of other liquid of density p. 



Taking the origin at the centre, let the shape of the common 

 surface at any instant be given by 



r = a + ? = + . sin (<rt + 6) ............... (1), 



where a is the mean radius, and S n is a surface-harmonic of order 

 n. The corresponding values of the velocity-potential will be, 

 at internal points, 



rrf'i t ltl 



and at external points * 



trfl n n + 1 



^-^iJsii.-.""(* + ) ............ (3), 



since these make 



d_ d<f>_ dfi 

 ~di~~~dr~ ~~dr' 



for r = a. The variable parts of the internal and external pressures 

 at the surface are then given by 



To find the sum of the curvatures we make use of the theorem 



* The argument here is that if we have a series of possible types of disturbance, 

 with time-factors e ait , e a<it , e" 3 *,..., where a 1 >a 2 >a 3 > ..., and if these be excited 

 simultaneously, the amplitude of the first will increase relatively to those of the 

 other components in the ratios a *~ a *& ) / a *~ a *^,. it . The component with the 

 greatest a will therefore ultimately predominate. 



The instability of a cylindrical jet surrounded by other fluid has been discussed 

 by Lord Bayleigh, "On the Instability of Cylindrical Fluid Surfaces," Phil. Mag., 

 Aug. 1892. For a jet of air in water the wave-length of maximum instability is 

 found to be 6-48 x 2a. 



t Lord Rayleigh, 1. c. ante p. 458; Webb, Mess, of Math., t. ix. p. 177 (1880). 



