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, 6 in the plane of a section, 

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

 Art. 63, 



r s 



(j) = A cos s6 . cos (at + e) (1), 



a 



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

 any instant will then be 



r = a+ (2), 



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



ord 



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 



- = -~- = o-A cos s6 . sin (at + e) (5). 



p d/t 



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 



11 1 d*r 



R~r ^dd 2 ' 

 or, in the notation of (2), 



Hence the surface condition 



p = T 1 /R + const., (7), 



gives, on substitution from (5), 



a , = 8(4 ,_l)^ (8)*. 



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

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

 Capillary Phenomena of Jets," Proc. Roy. Soc., May 5, 1879. The latter paper 

 contains a comparison of the theory with experiment. 



