225-226] INSTABILITY. 393 



applied to a thin membrane which is afterwards dissolved, the discontinuity 

 will persist, and, as we have seen, the height of the corrugations will continu- 

 ally increase. 



The above method, when applied to the case where the fluids are confined 

 between two rigid horizontal planes y -k, y=ti> leads to 



p(<r + kU)*cothkk+p'(<T+kU') 2 cothkh'=gk(p-p') ......... (iii), 



which is equivalent to Art. 224 (iii). 



226. We may next calculate the effect of an arbitrary, but 

 steady, application of pressure to the surface of a stream flowing 

 with uniform velocity c in the direction of x positive*. 



It is to be noted that, in the absence of dissipative forces, this 

 problem is to a certain extent indeterminate, for on any motion 

 satisfying the prescribed pressure-conditions we may superpose a 

 train of free waves, of arbitrary amplitude, whose length is such 

 that their velocity relative to the water is equal and opposite to 

 that of the stream, and which therefore maintain a fixed position 

 in space. 



To remove this indeterminateness we may suppose that the 

 deviation of any particle of the fluid from the state of uniform 

 flow is resisted by a force proportional to the relative velocity. 

 This law of resistance is not that which actually obtains in nature, 

 but it will serve to represent in a rough way the effect of small 

 dissipative forces ; and it has the great mathematical convenience 

 that it does not interfere with the irrotational character of the 

 motion. For if we write, in the equations of Art. 6, 



X = -p(u-c), Y=-g-/jiV, Z=-nw ......... (1), 



where the axis of y is supposed drawn vertically upwards, and c 

 denotes the velocity of the stream, the investigation of Art. 34, 

 when applied to a closed circuit, gives 



Q ............ (2), 



whence / (udx + vdy + wdz) = Ce~^ ...... ........... (3). 



* The first steps of the following investigation are adapted from a paper by 

 Lord Kayleigh, "On the Form of Standing Waves on the Surface of Eunning 

 Water," Proc. Lond. Math. Soc., t. xv., p. 69 (1883), being simplified by the 

 omission, for the present, of all reference to Capillarity. The definite integrals 

 involved are treated, however, in a somewhat more general manner, and the 

 discussion of the results necessarily follows a different course. 



