40 Proceedings of the Royal Irish Academy. 



where V"''"' = 0, and the vahies of v' are such that v satisfies (10), (11), 

 that V vanishes at the fixed bounding planes, and that v' initially vanishes 

 everywhere. Evidently in each layer v' is of the form 



v' = sinh l:i/ [F(t) cos fe +f(t) sin Ix} + cosh I;t/ { (p (t) cos Ix + T^(t) sin Ix} , (57) 

 where the functions of t have all to be determined. 



Equations (10), (11) give at each surface of separation four relations 

 among the functions and their differential coefficients with respect to time, 

 which correspond to the regions meeting there. The vanishing of v at the 

 fixed boundaries gives four other equations. There are thus obtained as 

 many equations as there are functions of t. (These equations differ from those 

 obtained in Lord Eayleigh's investigation* of the fundamental oscillations 

 by having, when all the unknown functions are brought to the left-hand 

 side, as their right-hand members given functions of the time instead of zero.) 

 The value of v', and therefore that of v, is evidently determinate ; and the 

 solution is unique ; for if v' + 'o' be substituted for x' , it appears that v" 

 must satisfy (10) and (11), must vanish at the boundaries, and be initially 

 zero everywhere, that is, it must represent a free oscillation which is initially 

 zero, and must therefore be zero always. 



Thus, in this case also, the analysis which has been given suffices to 

 include the most general disturbance possible. The complete determination 

 of V, even for an initial disturbance of the simple type discussed in the case 

 of uniform shearing, involves transcendental integrals. The expression for u 

 could be easily written down when that for v is obtained. 



ISTow, the form of the expressions for u, v shows that in this case also the 

 disturbance may increase very much. The first term in v will as before 

 increase very much if mjl is large ; and the hyperbolic functions in v show 

 that if I times the thickness of the layer is large, v' could neutralize this first 

 term in the neighbourhood of two planes only. It is not so clearly evident, 

 however, that, as in the simpler case, the disturbance may increase greatly, 

 even if / times the thickness of the layer is small, provided m times it is 

 large. It seems, however, reasonable to suppose that, if the initial wave- 

 length measured at right angles to the layer is small compared with the 

 thickness of the layer, the conditions of stability can depend little on the 

 conditions at the boundaries of the layer, and that therefore, in the cases in 

 which, as we have seen, the motion may be unstable when those boundaries 

 behave as fixed walls, it would also be unstable when the conditions to be 



^ " On tte Stability or Instability of certain Fluid Motions," i. andii. ; Proc. Lend. Math. Soc, 

 xi., xix. ; Collected Papers, i., iii. The functions of i in Lord Eayleigh's investigation are all 

 harmonic, and the elimination of their mutual ratios gives the equation determining the free periods. 



