50 



NA TURE 



[November 21, 1895 



sexagesimal minute of latitude and the hour ; the sailor's 

 knot as unit of speed can now replace the circumlocution 

 of sea miles per hour of p. 274, 449, and elsewhere ; but 

 the author avoids at least the landsman's solecism of 

 " knots an hour." 



Interesting applications of Bessel functions are dis- 

 cussed in the chapter relating to the propagation of the 

 tide in an estuary ; it may be mentioned here that the 

 figure for Jo( >-Jx) on p. 295 is easily realised by revolving 

 with appropriate speed the upper end of a vertical chain ; 

 this is the initial problem of Gray and Mathews's "Treatise 

 on Bessel Functions." 



Chapter ix. discusses the Surface Waves we are familiar 

 with in vessels of various shapes, embodying all the results 

 in this difficult subject which have so far been obtained. 

 Here is a branch of Hydrodynamics which will repay the 

 attention of young mathematicians, as every new problem 

 solved constitutes a distinct advance in the subject ; thus 

 the determination of a state of wave motion in a canal of 

 circular section, or in a spherical bowl, such as we see 

 every morning at the toilet, still await solution ; as 

 also the wave motion in the gutters during a shower, 

 when the viscosity takes a ruling part of the pheno- 

 menon. 



The investigations of the effect on waves of a capillary 

 film are easily extended to the case where the film 

 possesses a certain superficial density, as a flag or sail, 

 for instance, or 



" — the winning wave deserving note 

 In the tempestuous petticoat," — 



and where the film possesses flexural rigidity, as a 

 sheet of ice ; thus the results of Kelland, Kirchhoff, and 

 others on waves in canals are but slightly modified when 

 the free surface is supposed to be frozen. The minimum 

 wave velocity, where the ripples change into waves, is 

 now shown very elegantly by Mr. C. V. Boys on a 

 logarithmic diagram (NATURE, July 18, p. 273). 



Sir George Stokes's theory of Group Velocity of 

 Waves, introduced modestly in a Smith's Prize Question, 

 and developed subsequently by Osborne Reynolds, has 

 cleared up much of the mystery of the turbulence of 

 sea waves, and explained the reason of the sailor's adage 

 that every ninth wave is a big one, or every third wave in 

 crossing a bar ; for on this theory, if every ?«th wave is 

 a big one, the motion is principally due to the super- 

 position of two trams of waves, whose lengths are in the 

 ratio of (;« - i) to {in + i). 



The author summarises very clearly the researches of 

 Lord Kelvin and Lord Rayleigh on the wave-making due 

 to the motion of a ship, and the complicated pattern pro- 

 duced in the wake of a steamer, of a duck, or even of a 

 stick drawn through the water is illustrated in diagrams 

 on pp. 402, 403. If the unpublished investigation referred 

 to in the footnote of p. 403 relates to the explanation of 

 the curious appearance of the echelon bow waves which 

 form the fringe of the wake, where the wave fronts make 

 an angle with the keel, the tangent of which is double the 

 tangent of the angle between the wave fronts and the 

 general direction of the echelon, the following extract 

 from a letter by Lord Rayleigh will supply the 

 deficiency : — 



" A train of waves keeping up with a boat moving 

 NO. 1360, VOL. 53] 



along Ox with velocity Vq, the crests making an angle 9 

 with the keel, will be given by 



where 



a cos k (Yi - x sin fl - _y cos 1 



V„ sin e ; 



and the groups are due to the combination of two such 

 trains, defined by k, k + bk ; V, V + dV ; 6,6-\-d6% 

 subject to V«Xi oc;6-J, or k^N-" occosec'^ 6. 

 The resultant is thus given ultimately by 



2a cos kiyt - X %\r\.^ - y cos %) 



cos i {5(V/&y - .v9(Z' sin 6) - yd{k cos Q)\ j 



so that if the line of echelon makes an angle with the 

 keel, 



tan d) = ^^^ ^'" ^^ - 8(cosec 0) _ tan 



^{k cos fl) S(cosec"^e cos e) 2 + tan^ fl 

 or 



tan (e - (^) = i tan 6." 



(" Progressive Waves." By Lord Rayleigh. Proc. London 

 Math. Society^ vol. ix.) 



Chapter x. on Waves of Expansion discusses some 

 problems, such as oscillations of the atmosphere, but 

 passes over the applications which belong more properly 

 to the Theory of Sound. 



To account for many observed hydrodynamical pheno- 

 mena, notably of the passage of a ship through the water, 

 the hypothesis of perfect limpidity, postulated in Chapter i.^ 

 must be abandoned, and the equations of motion, such as 

 those employed by Osborne Reynolds in his investiga- 

 tions on turbulent motion and skin resistance, become 

 complicated to a formidable extent. It is, however, in 

 this direction that the most useful line of attack on new 

 and practical problems must be directed. 



The book concludes witu Chapter xii. on the Equili- 

 brium of rotating masses of liquid, such as the spheroid of 

 Maclaurin and the ellipsoid of Jacobi, and with a slight 

 sketch of Poincar^'s investigation of the secular stability, 

 and of the kindred researches of Bryan and Love. 



Considering that the author has already developed the 

 requisite Spherical Harmonic analysis, we think that a 

 resume of the theory of a Figure of the Earth would have 

 occupied very little space, and would have been very- 

 welcome, as Prof Adams's lectures on this subject are 

 not yet generally available. 



Prof Adams led up to Laplace's general case of strata, 

 varying continuously in density from one to the next, by 

 an examination of the particular cases of a spheroid 

 composed of two, three, or more strata of uniform 

 density ; so, too, the examination of the possibility of a 

 permanent distribution of the strata in the interior of an 

 ellipsoidal shell which is rotating about any fixed dia- 

 meter, is worthy of a careful examination by a young 

 mathematician as an interesting research ; the oscilla- 

 tions of such a shell have already been considered by 

 Mr. S. S. Hough {Proc. R. S., February 7, 1895). 



A comparison between the present treatise and Basset's 

 Hydrodynamics is interesting from the different modes 

 of presentation of parts of the theory ; we have reason 

 to be proud of both as representing a branch of mathe- 

 matical investigation in which foreign experts pay us 

 the compliment of conceding that we are capable of 

 teaching them something. A. G. Greenhill. 



