2 =m?S, (21). 



Stability of Viscous Fluid Motion. 619* 



The stability problem Las further been skilfully treated by 

 v. Mises * and by Hopf f , the latter of whom worked at the 

 suggestion of Sommerfeld, with the result of confirming the 

 conclusions of Kelvin and Orr. Doubtless the reasoning 

 employed was sufficient for the writers themselves, but the 

 statements of it put forward hardly carry conviction to the- 

 mere reader. The problem is indeed one of no ordinary 

 difficulty. It may, however, be simplified in one respect, as- 

 has been shown by v. Mises. It suffices to prove that q can 

 never be zero, inasmuch as it is certain that in some cases- 

 (/3=0) q is positive. 



In this direction it may be possible to go further. When 

 /3 = 0, it is easy to show that not merely q, but q — Irv. is 

 positive {. According to Hopf, this is true generally. Hence 

 it should suffice to omit k 2 — qjv in (18), and then to prove 

 that the S-solutions obtained from the equation so simplified 

 cannot satisfy (20). The functions S x and S 2 , satisfying the- 

 simplified equation 



drj 



where rj is real, being a linear function of y with real co- 

 efficients, could be completely tabulated by the combined use 

 of ascending and descending series, as explained by Stokes 

 in his paper of 1857 §. At the walls tj takes opposite signs. 

 Although a simpler demonstration is desirable, there can 

 remain (I suppose) little doubt but that the shearing motion 

 is stable for infinitesimal disturbances. It has not yet been 

 proved theoretically that the stability can fail for finite dis- 

 turbances on the supposition of perfectly smooth walls ; but 

 such failure seems probable. IVe know from the work 

 of Reynolds, Lorentz, and Orr that no failure of stability 

 can occur unless /3D 2 /v>177, where D is the distance 

 between the walls, so that /3D represents their relative motion. 



Terling Place, Witham, 

 August 21. 



* Festschrift II. Weber, Leipzig, 1912, p. 252 ; Jahresber. d. 

 Dcutschen Math. Ver. 13d. xxi. p. 241 (1913). The mathematics has a 

 very wide scope. 



t Ann. der Physik, Bd. xliv. p. 1 (1914). 



X Phil. Mag. xxxiv. p. 69 (1892) ; Scientific Papers, vol. iii. p. 583. 



§ Camb. Phil. Trans, vol. x. p. 106 ; Math, and Phys. Papers, vol. iv. 

 p. 77. This appears to have long preceded the work of Hankel. I may 

 perhaps pursue the line of inquiry here suggested. 



