460 SURFACE WAVES. [CHAP. IX 



To find the sum of the principal curvatures, we remark that, as an 

 obvious consequence of Euler's and Meunier's theorems on curva- 

 ture of surfaces, the curvature of any section differing infinitely 

 little from a principal normal section is, to the first order of small 

 quantities, the same as that of the principal section itself. It is 

 sufficient therefore in the present problem to calculate the curva- 

 tures of a transverse section of the cylinder, and of a section 

 through the axis. These are the principal sections in the 

 undisturbed state, and the principal sections of the deformed 

 surface will make infinitely small angles with them. For the 

 transverse section the formula (6) applies, whilst for the axial 

 section the curvature is d 2 %/dz* ; so that the required sum of 

 the principal curvatures is 



CL (TCL 



Also, at the surface, 



s (ka) cos s6 cos kz . sin (at + e) . . . (17). 

 p at 



The surface-condition Art. 244 (1) then gives 



For 5 > 0, <r 2 is positive ; but in the case (s = 0) of symmetry about 

 the axis a 2 will be negative if ka < 1 ; that is, the equilibrium is 

 unstable for disturbances whose wave-length (%7r/k) exceeds the 

 circumference of the jet. To ascertain the type of disturbance for 

 which the instability is greatest, we require to know the value of 

 ka which makes 



kaI Q ' (ka) 

 I 9 (ka) (k ( 1} ' 



a maximum. For this Lord Rayleigh finds & 2 a 2 = '4858, whence, 

 for the wave-length of maximum instability, 



%7r/k = 4-508 x 2a. 

 There is a tendency therefore to the production of bead-like 



