150 Lord Rayleigh on the Instability of a Cylinder 



for ^r 2 , 



^A( r A^^tl\ _^#J = k /2 -k 2 dyjr 2 m 

 r dr\ drr dr ) r dr r dr 



so that 



F=IM f¥!^l^p + 2i4ti±±il _»e *(<h+<K) . (22 ) 



L wr dr dr r J kr dr v 



The variable part of the capillary pressure is, as we have 

 already seen, 



Tg(ftV-l) 

 a 2 

 in which 



Thus, the condition to be satisfied when r = a is 



T(1-*V) jj, Jg=gffi + 2i4±\ -gf*fc. (23) 

 a'' na I zA:a ar «r r J ka dr K 



The forms of y{r h yjr 2 are to be determined by the equations 

 (15), (16), and by the conditions to be satisfied when r = 0. 

 It will be observed that fa satisfies the condition appropriate 

 to the stream function when there is a velocity-potential. 

 This would be of the form 



= ^J o (/&r), (24) 



so that 



fa=$(ru)dz= 3^; = r* fc Jo'v*). 



Thus 



f^ArJo^'/r) (25) 



is the most general form admissible, as may be verified by 

 differentiation. In this J (ik?*) satisfies the equation 



J w (^)+^J '(^) + J (^)=0. • • • (26) 



Since (16) differ from (15) only by the substitution of k r for 

 k, the general form for fa is 



^r 2 =BrJ '(^r) (27) 



By use of these values the first boundary condition (20) 

 becomes 



2k' 2 AJ / (ika)-h(k f2 + k 2 )BJo{ik'a) = 0. . . (28) 



We have next to introduce the same values into the second 

 boundary condition (23). In this 



^ = lifer A [j "(t*r) + ~£ J '(ttr)l = -AikaJ (zka) 



