Orr — Stability or Listahility of Motions of a Perfect Liquid. 19 



Neglecting, of course, terms of higher order than the first power of small 

 quantities, the condition for no slipping at the separating surface, obtained 

 by equating the two values of u at the surface in question and dividing by 



cos {nt + kx), becomes 



2h^Z + A {A - B) = 0. 

 In virtue of the relation 



dF/dt + {U+u) dF/dx + vdF/dy = 0,* 



we have {n + kU) h sin (nt + kx) + v = 0, 



and eliminating h, the result follows. 



In cases where d-JJ/dy'^ = 0, the substitution of (8) for (7) or the 

 equivalent supposition that the vorticity is unchanged,! constitutes a limita- 

 tion on the disturbance. In order to obtain a general solution we must 

 retain the factor n + kU in (7). For any value of y which makes 

 n + kU'=0 (8) need not be satisfied; and thus any value of - kU is an 

 admissible value of 7i satisfying all the conditions of the problem. Such a 

 solution involves slipping between layers whose separating surface in the 

 steady motion is given by the value of y referred to. 



Moreover, as this separating surface may equally well be a surface 

 separating layers of different rotation in the steady motion, we may have 

 solutions in which (11) is violated if n + kU = at the surface. If there 

 be no slipping at a separating surface for which n + kU = 0, equation (11), as 

 Lord Eayleigh points out, reduces to -y = 0. 



Lord Eayleigh then proceeds to consider the case in which d^ Ujdy^ is 

 not zero, and shows that if it be one-signed throughout, no complex value of n 

 can occur, and concludes that, if this condition be satisfied, the motion is 

 thoroughly stable. 



Lord Kelvin has argued+ that when, in (7), u + k U= 0, there is a " disturbing 

 infinity which vitiates the seeming proof of stability contained in Lord 

 Eayleigh's equations." 



I do not understand clearly what Lord Kelvin's objection really is ; possibly 

 he contends that where n-^- kU'=0, equation (7) when written in the form 



,2 ,,2 73 kvd^Ujdf' 



d^v dy^ - kH = ^^ (13 



' "^ n + kU ^ ' 



gives an infinite value for d-vlchf, or that the slipping to which Lord 

 Eayleigh's solution leads renders the motion unstable. 



'^ Lamb: "Hydrodynamics," §10. 



t Equation (8) is equivalent to d-vjdcc- + cl-vjdy'^ = or djdx {dvjdx — dujdy) = ; thus 

 we have dvjdx - dujdy =/(y, t), and this function of y, t, necessarily vanishes since the velocities 

 are periodic in x. 



X Phil. Mag., Sept., 1887, p. 275 ; Brit. Assor. Rep., 1880, p. 492. 



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