342 



NATURE 



[September 14, 191 1 



nent of flow along the curve at each element. If 

 the path be not closed the time-rate of growth of 

 the flow along any specified path which moves with 

 the fluid is obtained by subtracting the value of the 

 excess of the kinetic energy per unit mass of the fluid, 

 supposed incompressible and under no applied force, 

 above the pressure at the initial end of the curve of 

 integration, from the value at the terminal end. This 

 theorem, in its modification for a compressible fluid 

 in which the pressure is a function of the density, 

 and which is situated in a field of applied force, has 

 been truly said to comprehend the whole of the 

 dynamics of a perfect fluid. 



For the vortex-motion application the use is imme- 

 diate. If the curve be closed the time-rate of growth 

 of circulation for the moving circuit is zero; but the 

 integral which is thus shown to remain constant is 

 half the surface integral of the component of £, 17, C, 

 along the normal to each element of a surface of 

 which the curve is the bounding edge. Hence as the 

 fluid moves this surface-integral remains constant, and 

 if once zero is always zero. 



Thus the Kelvin theorem obtains the result for a 

 finite mass of the fluid that is obtained in another 

 way by Helmholtz, and it gives the subject a phy- 

 sical significance at once clear and convincing. The 

 theorem may, however, be made to give at once many 

 well-known results in ordinary fluid motion, which 

 are usually obtained by other methods. 



Almost every paper in volume iv. is of a kind 

 to arrest and hold the attention. The papers 

 entitled hydrokinetic solutions are particularly in- 

 teresting, both on account of their contents and from 

 the fact that they are an excellent example of the 

 best manner of the author, the manner of which the 

 pupils of his higher mathematical class had the most 

 frequent experience. It is probable, however, that 

 the letter to Tait on the influence of wind on waves 

 in water supposed frictionless, and on ripples, was 

 written off as soon as some observations of the kind 

 described in the letter had been made, probably in 

 the yacht Lalla Rookh, and had been interpreted by 

 the analysis written down immediately, and therefore 

 in more or less impromptu manner. Scott Russell 

 (though Lord Kelvin was not aware of the fact when 

 he wrote) had noticed, nearly thirty years before, 

 that surface tension must play the most important 

 part in the propagation of the very short waves often 

 seen on a smooth sheet of water ; but this short paper 

 proved that as the wave-length diminishes from a 

 great to a very small value, the speed of propagation 

 diminishes to a minimum for a certain wave-length, 

 and then continually increases. This wave-length 

 Lord Kelvin took as that of separation between ripples 

 and ordinary waves. 



Anyone can study the matter experimentally from 

 a boat drifting steadily forward in smooth water at a 

 speed of about half a mile per hour, while the wave 

 system is generated by a fishing line stretched by a 

 weight hanging in the water. How many physicists 

 have done this and taken photographs of the wave 

 patterns ("ripples in front and waves of the same 

 velocity behind ") with the excellent lenses and 

 NO. 2185, VOL. 87] 



cameras which are carried about on every holiday 

 excursion ? 



The paper on an alleged error in Laplace's theory 

 of the tides the present writer well remembers was 

 dictated in some little excitement by Lord Kelvin 

 just after he had read Ferrel's observations on 

 Laplace's process in the United States Coast Survey 

 Report for 1874, in which certain objections taken 

 by Airy in his article on tides and waves in the 

 Encyclopedia Metropolitana were quoted and ap- 

 proved. The point was a curious one, as to whether 

 a certain coefficient K 4 was or was not indeterminate 

 so far as the solution of the differential equation for 

 certain tides was concerned. It is now generally 

 admitted that the constant was not indeterminate, and 

 is correctly determined by Laplace's "exquisitely 

 subtle method." 



Perhaps a note might have been added, in connec- 

 tion with the short paper in volume iv. on the pre- 

 cessional motion of a liquid, on the two liquid 

 gyrostats which Lord Kelvin was so fond of show- 

 ing to visitors. In each water is enclosed within 

 a spheroidal shell, but in one the shell is oblate, in 

 the other prolate, with about 5 per cent, deviation 

 from sphericity in each case, and the shell rotates 

 about the axis of figure. The motion of the liquid 

 is stable in the first case, and the gyrostat takes 

 precessional motion like an ordinary gyrostat, while 

 in the other the motion is unstable. The first result 

 showed that the precessional motion of the earth does 

 not furnish a valid argument for the earth's rigidity. 

 The remarkable result has been established that the 

 motion of a liquid in a prolate case is stable if the 

 degree of prolateness be so great that the axial length 

 of the hollow filled by the water is more than three 

 times the diameter. As Greenhill has pointed out, this 

 fact is important in fixing the shape and size of the 

 gas-bags for a dirigible balloon. 



The most remarkable series of papers in the two 

 volumes are, however, those on waves in water. Here 

 the exponential solutions given by Fourier for the 

 linear conduction of heat are modified and applied to 

 the long chain of abstruse problems dealt with. The 

 discussion of the front and rear of a free procession of 

 waves, and the papers on the initiation and growth of 

 a train of waves, sheds much light on difficult ques- 

 tions regarding groups of waves, and the manner in 

 which they arise and are propagated from a given form. 

 In the hands of Mr. Green and others it is possible that 

 the analysis may have important applications in the 

 wave theory of light, in connection with Lord Ray- 

 leigh's theory of the origin of the periodicity of light 

 which has passed prisms and gratings and other 

 optical instruments. A detailed account of these 

 papers is a fit task for the specialist in hydro- 

 dynamics : that they should have been written by 

 Lord Kelvin during the last ten years of his life is 

 not the least wonderful thing about them. To those 

 who came in contact with him, it appeared that he 

 worked more slowly, but not less surely : the impetu- 

 osity of his genius was abated, not its natural force. 



OI the papers in volume v. we have said nothing, 

 and our space is exhausted. As a rule they are 



