14 Proceedings of the Royal Irish Academy 



quantitative comparisons ; but it is shown that here, too, Lord Rayleigh's 

 analysis suffices to inchide the most general disturbance, and that some 

 disturbances of initially simple type will increase very much. 



The chapter concludes with a brief consideration, in Art. 14, p. 41, of the case 

 in which, in the steady motion, the rate of shearing varies continuously from 

 one bounding-plane to the other, instead of by abrupt changes. Mathematical 

 difficulties render this portion of the discussion very unsatisfactory ; but 

 reasons are put forward for holding that at any rate if a disturbance has a 

 wave-length in the direction of flow which is sufficiently short, and has 

 initially one in the perpendicular direction which is much shorter, it will 

 increase very much (and afterwards diminish indefinitely). 



Chap. II., pp. 43-60, deals with flow ,in cylindrical strata, through a pipe 

 whose section is a circle, or an annulus between two concentric circles. 



Art. 15, p. 43, contains a brief account of Lord Eayleigh's discussion of 

 the fundamental free modes of disturbance. The only case in which he has 

 actually obtained the solution is that in which the law of velocity in the 

 steady motion is that appropriate to a viscous liquid in a complete circular 

 pipe, and then only for disturbances symmetrical about the axis. This is 

 the only law of flow, and this the only type of disturbance, which are at all 

 tractable ; and the remainder of the chapter is accordingly devoted to the 

 consideration of this particular problem. As in the case of plane strata, 

 discussed in Chapter I., each fundamental mode involves slipping in the 

 interior and, of course, at the boundaries. 



In Art. 16, p. 44, it is shown how any symmetrical disturbance may be 

 resolved into a system of Lord Rayleigh's type. 



And in Art. 17, p. 46, the result to which this leads is obtained directly 

 from the fundamental equations. 



In Art. 18, p. 47, the solution is written down for an initial disturbance of 

 type analytically simple, the radial velocity being sin m (r - h) sin kz, h being 

 the inner radius (which may be zero), and z being measured in the direction 

 of flow ; this solution is in terms of somewhat complicated integrals involving 

 Bessel functions of a purely imaginary argument. The approximate values 

 of these integrals, under certain conditions, are examined with a view to find 

 the magnitude of the disturbance at subsequent times ; and, in the definite 

 case in which the wave-length in the direction of flow is small compared with 

 the distance of the point considered from the axis, it is shown that if 

 the initial wave-length radially is still much smaller as a certain critical 

 time is approached, the disturbance increases in a very great ratio if the 

 point be not near a boundary. For any point, this critical time depends on 

 its distance from the axis. The justification which it has been thought 



