(>16 Lord Rayleigh on the 



At time t the greatest Talue of d-^rjdr occurs when 



r 2 = l-256x4^ . . (17) 



On the basis of his solution Oseen treats the problem of 

 the stability of the shearing motion between two parallel 

 planes and he arrives at the conclusion, in accordance with 

 Kelvin, that the motion is stable for infinitesimal disturbances. 

 For this purpose he considers " the specially unfavourable 

 case" where the distance between the planes is infinitely great. 

 I cannot see myself that Oseen has proved his point. It is 

 doubtless true that a great distance between the planes is 

 unfavourable to stability, but to arrive at a sure conclusion 

 there must be no limitation upon the character of the in- 

 finitesimal disturbance, whereas (as it appears to me) Oseen 

 assumes that the disturbance does not sensibly reach the 

 walls. The simultaneous evanescence at the walls of both 

 velocity-components of an otherwise sensible disturbance 

 would seem to be of the essence of the question. 



It may be added that Oseen is disposed to refer the in- 

 stability observed in practice not merely to the square of 

 the disturbance neglected in (6), but also to the inevitable 

 unevenness of the walls. 



We may perhaps convince ourselves that the infinitesimal 

 disturbances of (6), with U=/%, tend to die out by an 

 argument on the following lines, in which it may suffice to 

 consider the operation of a single wall. The argument 

 could, I think, be extended to both walls, but the statement 

 is more complicated. When there is but one wall, we may 

 as well fix ideas by supposing that the wall is at rest (at 



The difficulty of the problem arises largely from the 

 circumstance that the operation of the wall cannot be 

 imitated by the introduction of imaginary vorticities on the 

 further side, allowing the fluid to be treated as uninterrupted. 

 We may indeed in this way satisfy one of the necessary 

 conditions. Thus if corresponding to every real vorticity at 

 a point on the positive side we introduce the opposite vor- 

 ticity at the image of the point in the plane y = 0, we secure 

 the annulment in an unlimited fluid of the velocity-component 

 v parallel to y, but the component u, parallel to the flow, 

 remains finite. In order further to annul n, it is in general 

 necessary to introduce new vorticity at y = 0. The vorticities 

 on the positive side are not wholly arbitrary. 



Let us suppose that initially the only (additional) vorticity 

 in the interior of the fluid is at A, and that this vorticity is 



