the Stability of the Flow of Fluids, 67 



limit, the annulment of (17) is secured by the factor r a ; so 

 that complex values of n are still excluded, provided W l be 

 of unchangeable sign. In the present case the B of (19) 

 vanishes, and we have 



d?W = 2A ldW 



dyZ ~~ , r d% 

 so that (18) gives 



4 s 2 A 



^- — 2A, --xr— 2A5 



AV + **' 



satisfying the prescribed condition. 



The difficulty in reconciling calculation and experiment is 

 accordingly not to be explained by any peculiarity of the two- 

 dimensional motion to which calculation was first applied. 

 It may indeed be argued that the instabilities excluded are 

 only those of the exponential type, and that there may remain 

 others on the borderland of the form t cos t, &c. But if 

 the above calculations are really applicable to the limiting 

 case of a viscous fluid when the viscosity is infinitely small, 

 we should naturally expect to find that the smallest sen- 

 sible viscosity would convert the feebly unstable disturbance 

 into one distinctly stable, and if so the difficulty remains. 

 Speculations on such a subject in advance of definite argu- 

 ments are not worth much ; but the impression upon my 

 mind is that the motions calculated above for an absolutely 

 inviscid liquid may be found inapplicable to a viscid liquid 

 of vanishing viscosity, and that a more complete treatment 

 might even yet indicate instability, perhaps of a local cha- 

 racter, in the immediate neighbourhood of the walls, when the 

 viscosity is very small. 



It is on the basis of such a complete treatment, in which 

 the terms representing viscosity in the general equations are 

 retained, that Lord Kelvin f arrives at the conclusion that 

 the flow of viscous fluid between two parallel walls is fully 

 stable for infinitesimal disturbances, however small the amount 

 of the viscosity may be. Naturally, it is with diffidence that 

 I hesitate to follow so great an authority, but I must confess 

 that the argument does not appear to me demonstrative. No 

 attempt is made to determine whether in free disturbances of 

 the type e int (in his notation e mt ) 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 investigation 

 of forced vibrations of arbitrary (real) frequency, and the con- 

 clusion depends upon a tacit assumption that if these forced 



* Phil. Mag. Aug. and Sept. 1887. 

 F2 



