NATURE 



49 



THURSDAY, NOVEMBER 21, 1895. 



HYDROD YNAMICS. 

 Hydrodynamics. By Horace Lamb, M.A., F.R.S., Pro- 

 fessor of Mathematics in the Owens College, Man- 

 chester. (Cambridge : University Press, 1895.) 

 THE original edition of this work, published in 1879 

 under the title of a " Treatise on the Mathematical 

 Theory of the Motion of Fluids," gave the first impulse 

 to the cultivation in this country of the modern develop- 

 ments of Vortex and Cyclic Motion, with their Electro- 

 magnetic Analogues, the Discontinous Jets of Helmholtz 

 and Kirchhoff, the Dynamical Theory of the motion of 

 perforated solids through a liquid, and the examination, 

 as far as possible, of the effects of Viscosity. Previous 

 writers had confined themselves to simple applications 

 of the principle of Parallel Sections, to the bodily rotation 

 of liquid, especially of Ellipsoids, and to simple cases of 

 Wave Motion and of Tidal Phenomena. 



The analysis invented by Helmholtz for the special case 

 of a discontinuous jet has led, in Schwartz's hands, to 

 important developments in the Theory of Functions, 

 which still receive their most convincing explanation by 

 a return to the hydrodynamical analogue. A recent 

 article in the Math. Annalen, 1895, by R^thy, on this 

 subject deserves attention. So too, in the present 

 treatise, the author introduces much of the modern 

 Theory of Functions, guided by considerations of physical 

 interest ; but long analytical mvestigations, leading to 

 results which cannot be interpreted, have, as far as 

 possible, been avoided. 



The author quotes Poinsot's warning, " Gardons nous 

 croire qu'une science soit faite quand on I'a reduite k des 

 formules analytiques," as especially applicable to the 

 present branch of mathematical physics ; so he has made 

 the analytical results convincing and intelligible by 

 numerical illustrations, and by the insertion of a number 

 of diagrams, drawn carefully to scale and reduced by 

 photography. The hope expressed by the author that the 

 results of his numerical calculations will be found correct 

 is a natural feeling to those mathematicians who find by 

 experience that it is no small difficulty to make use of a 

 formula by turning its results into numbers. 



The so-called practical man can use a formula in this 

 manner, without understanding the theory upon which it 

 is based ; on the other hand the mathematical student is 

 f apt to imagine that, after the formula has been demon- 

 strated, it is an easy matter, hardly worthy of his attention, 

 to turn the formula to account in a numerical application, 

 until humiliating experience teaches him his error. 



The subject is opened out in the initial chapters in 

 much the same manner as in Basset's "Hydrodynamics." 

 The method of Conjugate Functions, now more commonly 

 called Harmonic Functions by continental writers, is a 

 powerful machine for the construction of results of motion 

 in two dimensions ; it is unfortunate that the method 

 cannot be extended to three dimensions, except when the 

 motion is symmetrical about an axis, when Stokes's stream 

 or current function presents a certain analogue. 



The hydrodynamical questions which can be solved by 

 NO. 1360, VOL. 53] 



the system of coordinates given by confocal quadrics are 

 very fully explained in Chapter v. ; for instance, the 

 motion of liquid due to the presence of an ellipsoid. A 

 very slight extension of the author's method enables us 

 to dispense with an infinite extent of liquid, and to 

 suppose it bounded externally by another confocal, useful 

 when the oscillations of gravitatmg liquid between two 

 confocal ellipsoids is to be considered ; so also the 

 author's own investigations in the preceding chapter of 

 the motion due to an infinite elliptic cylinder can be com- 

 pleted and amplified in a similar manner. We miss, too, 

 the special consideration of confocal paraboloids, where the 

 liquid necessarily extends to infinite distance ; also the dis- 

 cussion of the motion of the liquid filling a rectangular 

 box. Mr. Basset's method of exhibiting the result by two 

 symmetrical infinite series, due to Dr. Ferrers, can be 

 illustrated by supposing each series to represent the 

 effect of a shearing velocity in the shape of the box, the 

 superposition of the conjugate shear producing the same 

 effect as a rotation. 



Prof Lamb apparently does not approve of our insular 

 plan of collections of examples interspersed in the 

 Chapters ; but many important results, for which room 

 cannot be found without unduly swelling the book, can 

 in this manner receive mention, as in Basset's treatise. 



The plane motion of a solid through infinite liquid is 

 worked out completely, with accurate diagrams, in 

 Chapter vi., the error of supposing that the path of the 

 body can be looped when there is no circulation, is cor- 

 rected ; but we notice incidentally that, among all the 

 various functions employed by the author, the elliptic 

 functions are conspicuous by their absence. Kirchhoff s 

 equations for the general motion of a solid of revolution 

 in space filled with liquid afford appropriate applications 

 of elliptic functions, especially his curious equations 

 giving the position of the centre of the body ; but these 

 are not given in Chapter vi. 



It is always gratifying to be able to utilise some result 

 out of the vast accumulation of analysis written on 

 the elliptic functions ; attempts are just beginning at 

 the solution of new problems in Wave Motion em 

 bodying the functions ; for instance, by Willy Wien 

 in the Berlin Sitz. in discussing the effect of wind, and 

 by Korteweg and de Vries in the Phil. Mag., May 1895 

 in extending theorems concerning the solitary wave, 

 given (p. 420) by 



to the cnoidal wave 



h en 



t"\/'T^>''=\/rf 



reducing to the above when k — o. 



The author gives a complete though rather condensed 

 account of Vortex Motion in Chapter vii., excusing himself 

 on the ground that the recent investigations on this sub- 

 ject by Lord Kelvin, J. J. Thomson, Hicks, Larmor and 

 Love, derive most of their interest from their bearing on 

 kinetic theories of matter, theories which lie outside the 

 province of a treatise like the present. 



But Chapter viii. gives a complete and exhaustive in- 

 vestigation of Tidal Waves. In these cosmical problems 

 the sailor's units are most appropriate, the sea mile or 



