Ore — Stability or Instability of Motions of a Perfect Liquid. 63 



And the result obtained is that, if originally t// =f{,r) sin sO, then, at time t, 



2s{a'b''-a-'b')\p 



= {r'a-' - r-'a'} \ ip'h'' - p-'b')\pf'{p) +f{p) - s'-p-fip) ] sin s {6 - Vt/pjdp 



J 6 



+ ['}''b-' - r-'b') 



{p^a^ - p-^a-){pf\p)+f{p)-s^p-^f{p)] sin s{d-rt/p)dp, (7) 



the argument in V being p. 



And, again as before, this result may be otherwise obtained by noting 

 that, the second member on the left-hand side of (3) being evanescent, a 

 first integral of the equation is 



where F is a function determinate from the initial conditions, and in this 

 instance equal to f{r) sins(0 - Vt/r) ; then equation (8), integrated subject 

 to the conditions that xjj vanishes for r = a, r = b, will be found to lead to (7). 



Art. 24. Motion is Stable for a sv^ciently small Initial Disturbance varying 



as sin sO. 



As with the previous problems, if /(r) be any function of an ordinary 

 type, the motion is stable for the disturbance given initially by \p=f{r) siusd, 

 provided the initial value is small enough. For, if we denote the first integral 

 in the right-hand member of (7) by 



' UBins[Q - {C+C'p--)t\dp, 



b 



on integration by parts this may be written in the form 



d 

 h dp 



(2sC'tY' r^ Ur eosslO- [C-^C'r'\t] + {"IsC't)-' coss • {)- {C + C'p-'') t] -~ (p' U) dp, 



do 



(9) 

 the second term of which may be shown to be eventually of order Tl When 



the integral in the second term of the right-hand member of (7) is treated in 



a similar fashion, and the two terms combined, it is evident that in the 



resulting expression for ^ the terms of order t~^ cancel, and that ip is 



eventually of order t'"^. Thus, as is seen by differentiating ^, the radial 



velocity ultimately varies as t~~, and that in the direction of flow as t~'^. 



This argument applies even when a is increased indefinitely (in which 

 case C is zero, otherwise the velocity in steady motion would be infinitely 

 great at infinity). 



It does not apply, however, if C is zero, that is, if the fluid rotates like a 

 rigid body ; in this case (7) shows that the disturbance neither increases nor 



