458 SURFACE WAVES. [CHAP. IX 



direction of the axis of the jet does not affect the dynamics of the 

 question, and may be disregarded in the analytical treatment. 



We will take first the two-dimensional vibrations of the 

 column, the motion being supposed to be the same in each 

 section. Using polar coordinates r, in the plane of a section, 

 with the origin in the axis, we may write, in accordance with 

 Art. 63, 



r s 



(/&amp;gt; = J. cos s# . cos (oi + e) ............... (1), 



Cl&amp;gt; 



where a is the mean radius. The equation of the boundary at 

 any instant will then be 



r-o+f .............................. (2), 



where = -- cos s6 . sin (at + e) ............... (3), 



&amp;lt;T& 



the relation between the coefficients being determined by 



dt ~ dr 



for r = a. For the variable part of the pressure inside the column, 

 close to the surface, we have 



- = -~- = aA cos sO . sin (at + e) ............ (5). 



p ctt 



The curvature of a curve which differs infinitely little from a 

 circle having its centre at the origin is found by elementary 

 methods to be 



1_ = 1_ 1 &r 



R ~ r r* dQ* 

 or, in the notation of (2), 



Hence the surface condition 



p = T,/R + const., ..................... (7), 



gives, on substitution from (5), 



* For the original investigation, by the method of energy, see Lord Eayleigh, 

 &quot;On the Instability of Jets,&quot; Proc. Lond. Math. Soc., t. x., p. 4 (1878); &quot;On the 

 Capillary Phenomena of Jets,&quot; Proc. Boy. Soc., May 5, 1879. The latter paper 

 contains a comparison of the theory with experiment. 



