84 Proceedings of the Royal Irish Academy. 



displacements which are arbitrary functions of cc, z, t, indicating in a footnote 

 that this same method may be used as affording a complete discussion of the 

 former problem without any introduction of the v which satisfies (2), (3), (6). 

 He states that the essential convergence of these series proves that the steady 

 motion is stable, however small be v, provided that it is not zero. 



If V be zero, the series become divergent in a certain region, thus cci\incf 

 rise to the " disturbing infinity " alluded to in Part I., Chap. I., p. 19. 



Aet. -4. Lord RayhigTts Onticism of the latter Solution. 



Commenting on these investigations, Lord Eayleigh writes* — " ... I must 

 confess that the argument does not appear to me demonstrative. Xo attempt 

 is made to determine whether in free disturbances of the type c"'' (in his 

 notation e'"*) the imaginary part of n is finite, and if so whether it is positive 

 or negative. If I rightly understand it, the process consists in an investiga- 

 tion of forced \ibrations of arbitrary (real) frequency, and the conclusion 

 depends npon a tacit assumption that if these forced vibrations can be 

 expressed in a periodic form, the steady motion from which they are 

 deviations cannot be unstable. A very simple case suffices to prove that such 

 a principle cannot be admitted. The equation to the motion of the bob of a 

 pendulum situated near the highest point of its orbit is 



cl-oijclt- - rii~x = X, 

 where JTis an impressed force, li X = cosj^t, the corresponding part of cc is 



COSjJ^ 



jf + riv ' 

 Ijut this gives no indication of the inherent instability of the situation 

 expressed hy the free ' vibrations,' 



X = Ac'"^ + Bc'"'K" 

 This criticism is evidently directed against the argument in the second of 

 the two papers to which I have referred. 



Aet. 5. Zord Baykigh's Bemarks on the S^xcial Solution. 



In a later paper Lord Eayleigh, referring evidently to Lord Kelvin's first 

 investigation, wrotef : — 



"... In the particular case where the original vorticity is uniform, the 

 problem of small disturbances has been solved by Lord Kelvin, who shows 



* "On the question of the Stability of the Flow of Fluids," PhH. Mag., xxxiv., 1892, p. 67. 

 Colledted Papers, iii., p. 582. 



t "On the Stability or Instabilitj- of certaiu Fluid .Motions," Proc. Load. Math. Soc. xxvii., 

 1895 ; Collected Papers, iy., p. 209. 



