Ork — -Stahility or Instability of Motions of a Perfect Liquid. 47 



Now, if the equation 



__^P_,e.> = 0, (23) 



where P, Q are functions of r, have independent integrals, n = ^(r), u = xp{r], 

 the solution of 



dhc ^du ^ „ , , 



-^P-^Q^.Fir) (24) 



may be written 



'• 0(p)^(r)-^(p)0(r) 



If we apply this formula in the present instance, making use of the relation 



I\{x)K,{x) - I,{x)IC,{x) = 1/x, (26) 



and choosing the arbitrary constants in (25) so that u vanishes when r = a, 

 r = b, we again arrive at the equation (16). 



Art. 18. Application to Disturbance in which Initial Radial Velocity is 

 sin m {r - b) sin kz ; ivith suitable values of the Constants, it 

 increases greatly. 



Consider, then, a disturbance in which initially* 



u = Uf^ = sin m [r - b) sin kz, (27) 



and therefore 



w = Wo = (kr)'^ {sin m {r - b) + mr cos m (r - b) ] cos kz, 

 where sin m {a -b) = 0, 



this value of u being the coefficient of i in sin m [r - b) e^*\ 



Here {p (k^ + i)/(p) -f{f>) - pfip) ] .^"M--^) 



= [\p (k^ + ni^) + l/|o} sinm (p - b] - m cosm (p - 5)] e'^i~-'^^\ 



(28) 

 and accordingly (16) gives, selecting the coefficient of i, at time t, on changing 

 signs throughout, 



{I{ka)K,{kb) - I{kb)K,{ka)]u 

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



[-{p(F+m2)+/3-i} ^\nm{p-b)^r,i cos(p-&)] ^mk{z-Wt){I,{kp)K,{kh)-I{kh)K,{kp)'\dp 

 h 



+ { I{kr)K,{kb) - I{kb)K,{kr) } 



[- ( p{¥-irm^)^p~'^) smm{p-b)+7n cos (p-b)] sin k(z- Wt)[Ii{kp)Ki(ka)-Ii(ka)Ki(kp)]dp, 

 (29) 



* The example discussed in Arts. 20, 21 is analytically simpler. 



