Orr — Stability or Instability of Motions of a Perfect Liquid. 23 



should be unrestricted in magnitude or infinite in number (or, on the other 

 hand, to impose any restriction in either of these respects). The oscillations 

 characteristic of a compressible fluid, for instance, are propagated with one 

 unique velocity, or more properly with only two velocities equal in magni- 

 tude and opposite in sign. 



There is more force in the objection that Lord Rayleigh has not proved 

 that an arbitrary disturbance can be propagated in the manner he supposes. 

 He has, however, made out a lorima facie case. And a satisfactory investiga- 

 tion of the possibility of the expansion of an arbitrary function in a series of 

 given functions, as by Fourier's series, is generally a matter of difficulty. 

 In the case in which the stream is composed of layers of constant vorticity, 

 it may be proved that the requisite expansion is always possible, when the 

 arbitrary function is of an ordinary character. 



Aet. 3a. The Wave-Amplittide generally increases. 



There is, however, a circumstance connected with any fundamental 

 free disturbance which to some extent should modify Lord Eayleigh's conclu- 

 sion that for such a disturbance the steady motion is stable. Lord Rayleigh 

 has shown that, in a stream moving with uniform velocity, if a wavy surface 

 of discontinuity be created parallel to the direction of motion, and slipping 

 occurs in the direction of flow, the amplitude of the waves increases* (as 

 illustrated by the flapping of sails and flags). It may be shown that this is 

 the case also in each free disturbance of the fluid shearing. If, for simplicity, 

 the bounding planes be supposed at an infinite distance from the surface of 

 discontinuity taken to coincide (approximately) with the plane y =^ 0, we 

 have, on the positive side of this surface, 



V = Ae-^y sin h[x - Ut), 

 U+u = U+ 2Zy - Ae-^y cosk{x - Ut). 

 If y =f{x, t,) be the actual surface of separation (accurate to terms of the 

 first order), 



dfjdt + Udfldx = V = A sin Jc{x - Ut) ; 



and the general solution of this, which is of wave-length 27r/A; in x, is 

 f= At sin Jc {x - Ut) + C cos k{x ~ Ut) + C sin k{x- Ut). 

 This increase of amplitude, moreover, occurs in most of the more general 

 cases of flow discussed by Lord Rayleigh — in that alluded to in the conclud- 

 ing paragraph of Art. 1, if slipping he set up ; that of Art. 13, below; that of 

 Chap. II. ; that of Chap. iii. 



On the Instability of Jets," Proc. L. M. S., x., p. 4, 1879 ; Scientific Papers, t. i., p. 367 



