Okr — Stahility or Instability of Motions of a Viscous Liquid. 89 



disturbance. And although the function i) of equations (2), (7), (8) is not 

 easily obtainable in a form which enables us to calculate numerical values, 

 important conclusions may be drawn from the form which this solution gives 

 for V without any regard whatever to 'o. Whether the infinitesimal distur- 

 bances are stable or not, it furnishes, in fact, a proof that the motion may be 

 practically unstable, and shows qualitatively, and to some extent quantita- 

 tively, the circumstances in which instability may be expected. (In short, I 

 cannot make any substantial advance in the matter of showing that there 

 ^v^U be instability beyond pointing out what may be inferred from this 

 solution.) There is good reason for supposing that, if Ih, mh, lib are large, 

 the precise conditions which prevail at the boundaries cannot modify the 

 disturbance appreciably at any sensible distance, and thus cannot much affect 

 the question of stability for disturbances of small wave-lengths in the x and z 

 directions. It is seen that, if the viscosity is sufficiently small, just as when 

 it is altogether neglected,'^ the initial disturbance may, owing to the expres- 

 sion V' + {m - l^tf + n~ in the denominator of V, as given by (13), (14), 

 increase very much in spite of the exponential multiplier. We may, more- 

 over, easily amend the expression for v, by adding to it the proper solution 

 of the equation VH = 0, so as to obtain a solution which shall satisfy either 

 of the boundary-conditions v = ^, dvjdy = 0, but not both.f If we select 

 the former alternative, such a solution corresponding to an initial disturbance 

 in which 



V = Vq = B cos Ix sin my cos nz (28) 



is 



2v sinh \b _ Exp [ - vt (X^ + m^ - Im^t + r-^H^fo) \ 

 (A- + 7)1^) B cos nz~ . \^+ (m - l^tf 



X { sinh Xb sin [Ix + (m - l^t) y] - sinh X{b- y) sin Ix - sinh Xy sin \lx + (m - l(5t) b] 



Exp [- vt{X^ + m^ + hn^t + Z^/3^J^V3] 



P + {m + i^ty 



X { sinh Xh Qvii\lx-{m + l^t)y'\- sinhA (b - y) sin Ix - sinhA^/ sin[Za; - (m + l^t) b']], 



(29) 



in which A^ = /^ + n-. The solution in the case of a two-dimensioned dis- 

 turbance, in which n = 0, A = /, can be completed by writing down the 



* Compare Part I., Arts. 4-8, Avith Arts. 10, 11 here. 



t Of course, Lord Kelvin's typical initial distirrbance of (15) violates the boundary condition 

 dvjdy = ; the conditions v =^0, (P'vldy" = are somewhat simpler ; but even in that case I cannot 

 complete the solution in a form which gives results suitable for quantitative comparisons. 



