578 VISCOSITY. [CHAP, xi 



For small values of kh this makes o- 2 = -/'^u 2 , as in the case of absolute 

 discontinuity (Art. 225). For large values of M, on the other hand, <r= u, 

 indicating stability. Hence the question as to the stability for disturbances 

 of wave-length X depends on the ratio X/2A. The values of the function in 

 { } on the right-hand of (xiv) have been tabulated by Lord Rayleigh. It 

 appears that there is instability if X/2A>5, about ; and that the instability is 

 a maximum for X/2A = 8. 



2. In the papers referred to, Lord Rayleigh has further investigated 

 various cases of flow between parallel walls, with the view of throwing light 

 on the conditions of stability of linear motion in a pipe. The main result is 

 that if d 2 U/dy 2 does not change sign, in other words, if the curve with y as 

 abscissa and U as ordinate is of one curvature throughout, the motion is 

 stable. Since, however, the disturbed motion involves slipping at the walls, 

 it remains doubtful how far the conclusions apply to the question at present 

 under consideration, in which the condition of no slipping appears to be 

 fundamental. 



3. The substitution of (x) for (viii), when d 2 U/dy 2 =Q, is equivalent to 

 assuming that the rotation is the same as in the undisturbed motion ; since 

 on this hypothesis we have 



du' dv' .. , 



-j- =- J - = lyh?' (XV), 



dy da; 

 which, with (vi), leads to the equation in question. 



It is to be observed, however, that when d 2 U/dy 2 =Q, the equation (viii) 

 may be satisfied, for a particular value of y, by <r+kU=Q. For example, we 

 may suppose that at the plane y = a thin layer of (infinitely small) additional 

 vorticity is introduced. We then have, on the hypothesis that the fluid is 

 unlimited, 



*=A*** (xvi), 



the upper or the lower sign being taken according as y is positive or negative. 

 The condition (ix) is then satisfied by 



(xvii), 



( 



) 



where U denotes the value of U for y=0. Since the superposition of a 

 uniform velocity in the direction of x does not alter the problem, we may 

 suppose 7 = 0, and therefore o- = 0. The disturbed motion is steady ; in 

 other words, the original state of flow is (to the first order of small quantities) 

 neutral for a disturbance of this kind*. 



Lord Kelvin has attacked directly the very difficult problem of 

 determining the stability of laminar motion when viscosity is taken 



* Cf. Sir W. Thomson, " On a Disturbing Infinity in Lord Rayleigh's solution 

 for Waves in a plane Vortex Stratum," Brit. Ass. Rep., 1880. 



