92 Lord Rayleigh on [May 15, 



whence approximately 



a »= a ( 1 -S) < li? >- 



For the case n=0, (13) is replaced by 



a »=°( l -S) ^- 



We have now to calculate the area of the surface of (22), on which 

 the potential energy of displacement depends. We have 



s »'»'=.(I{ i+ ©' + (,w)T'' ! "- 



— -ff{\ + y$Kn cos 2 ?i0 sin 2 7^ + J» 2 a» 2 a~ 2 sin 2 w# cos s X$}r c£0 cZz 



= 2 { 27T<X + \irlfioit?a + ^7r^ 2 a ?J 2 a _1 } ; 



so that, if a denote the surface corresponding on the average to the 

 unit of length, 



a=27ra + l7ra- l (kW + nZ-l) x> ? . . . (15), 



the value of a being substituted from (13). 



The potential energy P, estimated per unit length, is therefore ex- 

 pressed by 



P=|^-iT(^ 2 a 2 + 7i 2 -l)^ 2 .... (16), 



T being the superficial tension. 



For the case « = 0, (16) is replaced by 



F=±7ra- l T(kW-l)* z (17). 



From (16) it appears that, when n is unity or any greater integer, 

 the value of P is positive, showing that, for all displacements of 

 these kinds, the original equilibrium is stable. For the case of dis- 

 placements symmetrical about the axis, we see from (17) that the 

 equilibrium is stable or unstable according as ka is greater or less than 

 unity, i.e., according as the wave-length (27rk~ l ) is less or greater 

 than the circumference of the cylinder. 



If the expression for r in (12) involve a number of terms with 

 various values of n and Jc, the corresponding expression for P is found 

 by simple addition of the expressions relating to the component 

 terms, and contains only the squares (and not the products) of the 

 quantities «. 



