Orr — Stahiliti/ or InsiaUlitij of Motions of a Perfect Liquid. 41 



satisfied at them are those which prevail when the stream is disturbed 

 through its entire thickness. If this argument is legitimate, even the brief 

 discussion of the forms of u, v which has been given might be dispensed with. 



Aet. 14. The Case of Continuous, hut Varying, Vorticity-. 



The explanation just given of the possibility of instability in the case 

 of a finite number of layers cannot, at least prima facie, by making the 

 number of layers infinite, be extended to cover the case of continuously 

 varying vorticity. Por, as presented above, it requires at least that the 

 original wave4ength at right angles to the stream should be small compared 

 with the thickness of some layer. 



In this general case, confining ourselves to two dimensions and using the 

 stream-function t//, we readily obtain instead of (31) the more general equation 



^ rr ^ \ o . d-djcPU ^ 



This equation is intractable, and, as has been seen, the consideration of 

 disturbances alona which vary as e'"^ is not.&ufficient; but some light may 

 be thrown on the question under discussion by considering a certain type of 

 approximate solution. The approximate solution of a differential equation 

 when an accurate one is not feasible is, however, a question of considerable 

 delicacy. Let us endeavour to see under what conditions this equation 

 would be satisfied by the approximate value 



c os{^(a;- m) + my} ' 



^ • l" + {m- ItdUldyf ' ^ ■> 



which of course implies regarding dUldy as a constant. One might be 

 disposed to state that the necessary conditions are that the terms neglected 

 should be small compared with those which are retained either in didt . v Y? 



d 

 or in Z7~-. v^^; i.e. with lU. Evidently, however, the addition or sub- 

 traction of a constant to or from TT should leave the problem unaltered (or 

 at most require only some modification of the end-conditions).* In any 

 equation indeed, algebraic or differential, the division into terms is to some 

 extent a matter of convenience ; and if we strike out a term, it is not quite 



* It seems evident that some consideration of end-conditions in all these problems is desirable, if 

 we reflect that by ignoring them we might reduce the question of the stability of a stream of uniform 

 velocity to that of a liquid at rest. These questions are, it seems obvious, piactically diffeient. 

 R. I. A. PROC, VOL. XXVII., SECT. A. [6j 



