80 Proceedings of the Royal Irish Academy. 



CHAPTEE I. 



LoED Kelvin's Investigations, especially the case of a Stkeam which 



IS Sheaeing Unifoemly. 



Aet. 1. The ProUems which Lord Kelvin discussed. - 



The stability or instability of steady laminar motion, when viscosity is 

 taken into account in the disturbed motion, has been discussed by Lord 

 Kelvin for two cases. One of these is that of a fluid undergoing simple 

 shear, the problem which, when viscosity is ignored, formed the chief subject 

 of Part I., Chap. I., of the present paper ;* in the other, the steady velocity is 

 a quadratic function of the distance from a plane boundary, as with a viscous 

 fluid v/hich is moved between two fixed parallel infinite planes by gravity or 

 by applied pressure. 



As somewhat subtle controversial matters are to be touched on in what 

 follows, it appears desirable, with a view to facilitate the reader's compre- 

 hension of the points at issue, to give to some extent an outline of the 

 substance of his investigation. 



Lord Kelvin, in one paper,t discussed the former of the two problems 

 alluded to ; in another,^ he attacked the latter problem on somewhat 

 different lines, and in a foot-note indicated that this method applies equally 

 to the former, and thus constitutes a second solution of it. It will be 

 convenient to allude to the former solution as his "special" solution. 



Aet. 2. Abstract of his Sijccial Solution in the case of the Stream 

 shearing uniformly. 



Eeferring, then, to his first paper, if we denote the plane boundaries by 

 2/ = 0, y = 1), suppose that the former is reduced to rest, that the velocity in 

 the steady motion is U = [5y, and that in the disturbed U + v.-, v, w, and 



* Proc. R.I. A., xxvii., A. No. 2, p. 9. 



t " Stability of Fluid Motion^RectUinear Motion of Viscous Fluid between two Tarallel Planes," 

 Phil. Mag., Aug. 1887. 



X "Stability of Motion— Broad River flowing doun an Inclined Plane Bed," Pliil. Mag., Sept., 

 X8S7. 



