54 l^roceedings of the Royal Irish Academy. 



And it may Idg shown that the second term in this is eventually equal to 



"1 d (U\ 



^ ^m{kz-kt{C-r^C')] 



.P dp \P J- 



(62) 



4.kH'G^ 



Now when the second integral in (29) is similarly treated, and both terms of 

 (29) are combined, it will be seen that the terms which vary inversely as t 

 cancel each other, and that the value of u thus eventually varies as t'-. But 

 the equation of continuity shows that, this being so, the value of lo eventually 

 varies as t~^. 



Thus for a disturbance of the type cited — and the argument is seen to 

 apply equally to any ordinary disturbance which is periodic in the direction 

 of flow, and symmetrical round the axis — the steady motion is stable, irrovided 

 the initial disturlance is small enough, the kinetic energy of the relative 

 motion eventually varying inversely as the square of the time; and this is 

 true whatever the values of m, h, a, l. 



Art. 20. Disturbance in which Initicd Radial Velocity is sin ?/r(r- - If) mikz ; 

 vjith suitahle values of the Constants it Increases Greatly. 



As another example, consider a disturbance in which initially 

 u = u,-, = sin rii^ (r- - If) sin kz, \ 



w = vj^ = (At)-^ (sin riv (r- - ¥) + 'Invr cos nf (r^ - If)] cos kz ) 

 where sin nr {a} - If) = 0. 

 Here 



-p{kHi!pY{p>f{p)^pf\p>-i^-''iy+i^'p+p-')^^^^-',i\p^^^^^ 



(64) 

 and accordingly the right-hand member of this equation is to replace the 

 first factor in the integrals of (29). Suppose riut large, and let us examine 

 the value which this modified form of (29) gives for u. 



Consider first the former of the two integrals, viz., that whose range is 

 from 6 to r. It may be written as the difference of two, thus : — 



1 1 [(4/nV' + k-p + p-') cos (y/r (p- - &-) + kz - kt {Cp- + C')] 



+ 4//r|o sin | //?'(/-/ - If) + kz - kt{Cp' + 6"))] 

 X [I (^kp) K, {kh) - I {kb) K, {kp)-] dp 



[(4r/tV' + k-p + p-') cos |m^(p^ - b') - kz + kf(Cy- + C')} 



+ 4:m^p sin 1 nf (p- - ¥) - kz + kt (Cp- + C) j ] 

 X [/, (kp) K, (kb) - I (kb) K, (kp)-] dp. (65j 



We will concern ourselves only with a time at which nf - kCt = 0. Taking 



