RECENT PBOftRESS IN HYDRODYNAMICS. 'Jl 



would not be felt. Experimentally the subject has been very fully 

 investigated,' but no attempt lias been made towards a general theoretical 

 investigation of the stability of such motions. Lord Rayleigh has, 

 indeed, considered some general aspects of the qaestion in two important 

 papers, in the ' Proceedings of the London Mathematical Society, '^ some 

 of the results of which it may be well to state here. In the first, he 

 shows that, in the case of cylindrical columns, the disturbing cause due 

 to surface-tension lias most efiect when, for harmonic disturbances, the 

 ratio of wave-length to diameter is about 4-508 ; and also determines the 

 rate of falling away from steadiness for small harmonic disturbances in 

 the cases of plane and cylindrical sheets of discontinuity. The results 

 lead to the supposition, as is pointed out in the second paper, that the 

 sensitiveness of sensitive jets would increase indefinitely with the pitch, 

 which is not, in general, true. The explanation he finds in the operation 

 of the viscosity of the fluid, which, for water, is found to be such that, if 

 a plane surface of discontinuity existed at any moment, then, after the 

 lapse of one second, there would be a layer of transition, consisting of 

 vortex motion, of a thickness of half a centimeter. In this paper the 

 fluid is treated as frictionless, but the jets as containing vortex motion, 

 and in two dimensions. For a layer of given thickness, with velocities 

 V and —V on the two sides, and uniform vorticity between, the motion is 

 unstable when the wave-length of disturbance is great in comparison 

 with the breadth, and stable when the wave-length is small. For a jet 

 in fluid, at rest, with its centre moving with velocity V, and velocity 

 decreasing uniformly to zero on either side, the motion is stable for 

 symmetrical disturbances, or when the wave-length is small compared 

 with the breadth of the jet, and unstable when it is great. In general, 

 in a case where the velocity increases continually or decreases continually 

 between the fixed boundaries of the fluid, a jet will be stable. 



Another kind of discontinuous motion, viz., the propagation of a 

 shock through a slightly compressible liquid,, has been treated by 

 Christoffel,3 in a manner founded on that of Riemann. In such a case 

 there will be a surface of discontinuity in the fluid, which moves forward, 

 and such that the pressure, density, and velocity are different on the 

 two sides. If w be the normal velocity of progression of a point of the 

 surface, ii,, pj, tig? P2> ^^^ normal velocities and densities of the fluid on 

 the two sides, then the equation of continuity is 



Pi (S2i — w) = p.2 (£2o — (•)), 



and it is easily shown that the dynamical equations are three of the 

 form — 



P2(^2 ~ i>>)(ui — ^l■2) + (i'l — P-i) cos o = 0, 



where ii.v.w are components of 12. o, /3, y its direction and p the pressure. 



' See chiefly Savart, Ann. de Chimie et de Physique, liv. Iv. ; and Magnus, Fogg. 

 Ann., Ixxx. xcv. cvi. ; also PJiil. Mag. (4) i. A very full historical account is given 

 by Plateau, in his Statique expcrimentale et tMorique des Uquides, tome ii. ch. xii. 

 to which the reader is referred. Since the appearaaice of this, Oberbeck has pubUshed 

 a paper on the same subject, in Wied. Ami. ii. p. 1 (1877) ; and Ridout, Nature, xviii. 

 p. 60i. See also Enctjc. Brit. Art Hydraulics by W. C. Unwin, 



" ' On the Instability of Jets,' Proc. Land. Math. Soo., x. p. 4 ; 'On the stability 

 or instability of certain fluid motions,' lb. p. 57. 



^ ' Untersuchungen uber die mit dam Fortbestehen linearer partieller Difierential- 

 gleichungen vertraglichen Unstetigkeiten.' Brioschi, Annali di Matematica, viii. 

 p. 81. 



