494 Mr. W. J. Harrison on the Stability of 



which gradually decreases, and is finally exchanged for 

 actual stability " *. 



I have given below three methods of solving the equations 

 of motion, of which only one has been employed. The 

 other two are too cumbrous for practical use, though more 

 rigorous. There still remains another method, which has 

 the advantage of not needing the same assumptions for the 

 purposes of approximation, and which is especially adapted 

 for very viscous liquids, but can only be employed for 

 disturbances of great wave-length. However, as the stability 

 of the motion is determined by its stability for great wave- 

 lengths, this method will furnish the precise information we 

 need. I hope to develop this solution in a future paper. 



§2. "We shall confine the problem to two dimensions and 

 take the origin of coordinates (x, y) in the undisturbed 

 interface, the axis of x being in the direction of flow, the 

 axis of y vertically upwards. 



The equation which is satisfied by the stream function M* 

 can easily be shown to be 



(•*■-&*—(£ s-S|)^ ■ w 



where v is the kinematical coefficient of viscosity. 



We shall take the undisturbed motion of the lower liquid 

 to be given by 



This implies an infinite velocity at y = — co , but as we are 

 only concerned with the condition near the interface this 

 need not cause any serious trouble. Lord Rayleigh makes 

 the same assumption in his work on motion past a corrugated 

 wall f . 



For the upper liquid 



t „' = w y + joy, 



where we must have 



B = B', 

 v P Q = v'p'W, 



by reason of the continuity of velocity and traction across 

 the interface. 



* Scientific Papers, vol. i. p. 480 ; Proc. Lond. Math. Soc, xi. p. 63, 

 1880. 

 t Scientific Papers, vol. iv. p. 89 ; Phil. Mag. xxxvi. p. 368 (1893). 



