1,914] ON THE STABILITY OF VISCOUS FLUID MOTION 275
where S1} S2 are any two independent solutions of (18), and the integrations are extended over the interval between the walls. An equivalent equation was given a little later (1908) independently by Sommerfeld*.
Stability requires that for no value of k shall any of the q'a determined by (20) be negative. In his discussion Or arrives at the conclusion that this condition is satisfied, though he does not claim that his method is rigorous. Another of Orr's results may be mentioned here. He shows that p + kfiy necessarily changes sign in the interval between the walls.
The stability problem has further been skilfully treated by v. Misesf and by Hopf J, 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 (# = 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 — h*v, is positive§. According to Hopf, this is true generally. Hence it should suffice to omit 7c2 — qjv in (18), and then to prove that the ^-solutions obtained from the equation so simplified cannot satisfy (20). The functions Sj, and $2J satisfying the simplified equation
where 77 is realt being a linear function of y with real coefficients, could be completely tabulated by the combined use of ascending and descending aories, aa explained by Stokes in his paper of 1857 1|. At the walls rj 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 disturbances on the supposition of perfectly smooth walls ; but such failure seems probable. We know from the work of Keynolds, Lorentz, and Orr that no failure of stability can occur unless @D*jv > 177, where D is the distance between, the walls, so that j3D represents their relative motion.
* Atti del IV. Gongr. -intern, dei Math. Eoraa (1909).
I Festschrift H. Weber, Leipzig (1912), p. 252; Jahresber. d. Deutschen Math. Ver. Bd. xxi. p. 241 (1918). The mathematics has a very wide scope.
$ Ann. der Physik, Bd. XLIV. p. 1 (1914).
§ Phil. Mag. Vol. xxxiv. p. 69 (1892) ; Scientific Papers, Vol. m. p. 583.
|| Camb. Phil. Tram. 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