Ore — Stability or Instability of Motions of a Perfect Liquid. 35 



and (16) thus assumes the form 



A sinh \h f{y) = sinh \(b - y) sinh \rj {\^f{ri) -/" (rj) } dr) 



+ smhXy\ sinhA(&-r,){X^/(»,)-/"(r,)}f6); (22) 

 Jy 



and. it may now be directly verified" that this result is true, provided /(?/) and 

 /' (y) are finite, continuous, and differentiable between and h, and/(0) and 

 f{h) both vanish, as is the case in the problem to which the theorem is to be 

 applied. If /(O), f(h) are not zero, we require to add to the right-hand member 



A/(0) sinh \ {h - y) - A/(&) sinh \y; 

 and even this apparent exception may be made to conform to (16), if we agree 

 to consider that ^ (t?) becomes infinite at the limits 0, b, in such a fashion 

 that for infinitesimal ranges dr\ at the lower limit ^{r\)dr) =/(0)/sinh \b, and 

 at the upper, ^ (rj) dt) = -/(6)/sinh Xb. 



If, then, there be an initial disturbance in which v =f{y) e-'^, its value at 

 the time t as thus obtained is given by 



\Qm\i\b.v = ^mh\[b - y){\mh\n{\' f[ri) - f {n)]e^'hkLEIldri 



J 



+ sinhA3/| sinh A (5 - r,){AV(r?) -/'(r?)}e'''^'"-^')^»?, (23) 



in which Z7 is a linear function of tj. 



This is the solution on the supposition that the disturbance continues to 

 have a wave-length in the x direction equal to 27r/A.* 



The discussion may be made more general by extending its scope so as to 

 include three-dimensional disturbances at least as far as finding the value of v. 

 We may, without loss of generality, suppose that one of the bounding planes 

 is reduced to rest, as any other case may be obtained from this by replacing 

 xhy X - d, where c is constant. Let then the velocity in the steady motion 

 be given by ?7 = By. Consider the propagation of the disturbance in which 

 the initial values of w, v, w are 



Uq = A sin Ix cos my cos nz, 

 Vq = B cos Ix sin my cos nz, 



Wq - G cos Ix cos my sin nz, (24) 



where sin mb is zero, and, as follows from the equation of continuity, 



lA + mB + nG = 0. (25) 



In each " free " disturbance, v, as a function of x, y, t, is to be taken to vary 

 as sinh \y e^'^(^-^')^) on one side of the plane of discontinuity y = i], and as 

 sinh X(b - y) e«'^(^-^''^) on the other, where now 



A- = 1' + n\ (26) 



* The solution can be made to satisfy definite assigned end-conditions : see below, Arts. 6, 7. 

 R. I. A. PROC, VOL. XXVII., SECT. A, [4] 



