148 Lord Rayleigh on the Instability of a Cylinder 

 lf>' = 0, 



/• \2— ± V- 1 - — tea- ikuoq nry . 



T (1-&V) 

 pa? J 



as found in the former paper. In this expression ika J 7 J is 

 a real positive quantity for all (real) values of ka ; so that the 

 displacement is exponentially unstable if ka < 1, and periodic 

 if ka>l, as was to be expected. In the former case the 

 values of in are numerically greatest when ka = ir/4:'5. 



In the other extreme case where inertia may be neglected 

 in comparison with viscositv, we have 



^« (id 



pa 3 p'/p.Jo 



so that the instability is greatest when ka has the same value 

 as in the first case. 



The general form of the quadratic is 



(m) 2 H-m./^7 / )H-H(Pa 2 -l)=0, . . . (12) 



where H is positive. 



If ka < 1, both values of in are real, one being positive and 

 the other negative. The displacement is accordingly unstable, 

 and the greatest instability occurs with the above-defined 

 value of ka. If, on the other hand, ka > 1, the values of in 

 may be either real or imaginary. In the former case both 

 values are negative, and in the latter the real parts are nega- 

 tive, so that the deformations are stable. 



The investigation applicable to a real viscous liquid of vis- 

 cosity fi, or pv, is much more complicated than the foregoing, 

 mainly in consequence of the non-existence of a velocity- 

 potential. But inasmuch as the motion is still supposed to be 

 symmetrical about the axis, the equation of continuity gives 



= 1 difr __!^ /-ion 



r dz } ~ r dr' *•-'•• v y 



where yjr is Stokes's current function. For small motions i/r 

 satisfies the equation* 



\dr 2 rdr^ dz 2 v dt)\dr 2 r dr + dz 2 ) r ~ V ' ' l *> 



In the present question yfr as a function of z and t is propor- 

 tional to e^ nt+ke \ and it may be separated into two parts, ^ 



* Camb. Trans. 1850. See also Basset's ' Hydrodynamics,' vol. ii. 

 p. 259. 



