Ork — Stahility or InstaUlily of Motions of a Perfect Liquid. 45 



symmetrical about the axis, and to examine the propagation of one initially 

 of type analytically simple. In the more general case of an annular tube, 

 whose outer and inner radii are a, l, the problem in expansions may evidently 

 be reduced to the following : — 



Given f{r) an arbitrary function of r, find a function ^ such that for 

 values of t between I and a 



f{r) = \\{p)F,{r)dp, (9) 



where Fp{r) is a given function of r, defined thus: when r lies between 

 h and p 



F^r) = A {I,{hr) K,(kh) - I,{M)) K,(kr)}, (10) 



and when r lies between p and a 



F,{t) = B[I,{kr) K,{ka) - I,(ka) K,{kr)], (11) 



Ki(kr) denoting the solution of (8) which vanishes when r is infinite, k being 



a given constant, and the values of A, B being so connected that the two 



forms of Fp {r) have identical values at their common limit, r = p. 



As there is an arbitrary multiplier in Ki(kr), and as any convenient 



values of A, B may be chosen, we will take Ki{kr) to be that solution of (8) 



which, for a large real positive value of h\ approximates to tt^ {2kr)"^ e~^^' ; 



and we will take 



A = I,{kp) K,{ka) - I,{ka) K,{kp), (12) 



B = I,{kp) K,(kh) - I,{kV) K,{kp). (13) 



The form of 0(|o) may be discovered by a procedure similar to that of 

 Art. 4, and found to be given by the equation 



- <i>{p) [hka)K,{kh) - I,(kh) K,{ka)\ = p (k^ + -^ f{p) - f(p) - pf'(p). 



^ (14) 



The equation expressing the expansion, if any such be possible, is thus 



\^p{jc^^-^f{p) -f{p) - p/Xp)^ Fp{r) dp, 



- {I,{ka)K,(kh) - I,{kJ))X,(ka)}f(r) = 



= {I,(kr) K,(ka) - I,(ka) K,{kr)} 



X 'Uk^ + -)jf(p) -f(p) - pfip)]^ imp) K, ifl) - I,{kh) K,{kp)-] dp, 



+ {I,{kr) K,(kh) - I,(kb) K,(kr)] 



• X ''^\p{^^'+-}jfip)-f(p)-pr(p)\ [mp)Kfka)-f{ka)K,{kp)-\dp- 

 ^ (15) 



and it may be easily verified that this is true, provided that /(?•) and f (r) 

 are finite, continuous, and differentiable throughout the region, and that f(r) 

 vanishes at both the boundaries r = a, r = h. The most general symmetrical 

 disturbance can thus be analysed into elementary ones of Lord Eayleigh's type. 



