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542 NA TURE [October 3, 1895 erosion and denudation have modified the primary features on a gigantic scale f and a valley so deep as the northern part of Lake Baikal is, has been dug out across the former direction of the chains. The lake is thus an immense erosion valley which only partially has been determined by the structural valleys at the foot of the plateau, but has received its final shape through erosion, which made several parallel lakes coalesce as the moun¬ tains once separating them were pierced through and obliterated. This instance will already give an idea of the interest which attaches to the volume now published, and the wealth of data which will be found in it. We sincerely desire, in the interests of geography, that at least these new volumes of the series should be rendered accessible to West European geographers. The described region is very thinly populated, and contains but few explored remains of the past. As to its flora, it has been properly explored only on the Olkhon Island. The little, however, which is known in these two 'directions is well summed up, and will give a sound basis for ulterior exploration. We hope to find in the forth¬ coming volume a summary of all that is known about the fauna of the lake. P. K. APPLICATIONS OF BESSEL FUNCTIONS. A Trcali.sc on Bessel Functions and their Applications to Physics. By Andrew Gray, M.A., and G. B. Mathews, M.A. (London : Macmillan and Co., 1895.) HIS book, like the kindred work of Prof. Byerly on “ Fourier’s Series and Spherical Harmonics,” marks the modern system of mathematical treatment, and may be contrasted with Dr. Todhunter’s “ Functions of La¬ place, Lame, and Bessel,” of twenty years ago. At that time it was considered desirable to develop the purely mathematical analysis quite apart from the physical considerations to which it owed its life and interest ; keeping the pure and the mixed mathematics in separate water-tight compartments, so to speak, with an im¬ penetrable bulkhead between. But as the Bessel function, like every other function, first presented itself in connection with physical in¬ vestigations, the authors have done well to begin, on p. t, with a brief account of three independent problems which lead to its introduction into analysis, before enter¬ ing upon the discussion of the properties of the Bessel functions. These three problems are : the small oscillations of a vertical chain, the conduction of heat in a solid cylinder, and the complete solution of Kepler’s problem by ex¬ pressing radius vector, true and excentric anomaly in terms of the mean anomaly. It is very extraordinary that Kepler’s problem should, as a general rule, be still left unfinished in the ordinary treatises, considering that the Bessel function is implicitly defined in the equation ; but we need go back only txventy-five years, and we find Boole’s “ Differential Equations ” ignoring the Bessel Function and the solution of the general Riccation equation which it provides. In those days it was customary to speak of any solution, not immediately expressible by algebraical or trigonometrical NO. 1353, VOL. 52] functions, as “ not integrable in finite terms ” ; an elliptic integral was skirted round with the remark that it was “ reducible to a matter of mere quadrature,” and even the homely hyberbolic functions were tabooed. String is the favourite material of the mathematician for illustrating catenary properties ; but it is a relief to find that the authors have provided a chain for the discus¬ sion of the oscillations when suspended in a vertical line. The banal word string turns up accidentally two or three lines lower down (line 10, p. 1), but if a piece of string is used by the side of a length of fine chain, such as is now purchasable, the unsuitability of the string, by reason of its lack of flexibilty and its kirikiness, for the representation of catenaries and their oscillations, is at once manifest. The small plane oscillations of the chain about its mean vertical position are of exactly the same character as the slight deviations from the straight line due to spinning the chain from its highest point of suspension; and this procedure has the advantage of showing a per¬ manent figure, similar to that given for J 0 ( Jx) on p. 295 of Lamb’s “ Hydrodynamics ” ; with a little practice the knack of producing one, two, three or more nodes at will is easily attained. Thus with a piece of chain 4 feet long, the number of revolutions per second should be 0-54, 1-24, 1-95, 2-65, &c. The Bessel function was first introduced by the in¬ ventor for the complete solution of Kepler’s problem, namely, to express the variable quantities in undisturbed planetary motion in terms of the time or mean anomaly p = lit -j- e — TXT. The authors avoid the awkward integration by parts employed by Todhunter in determining the excentric anomaly 0 by means of a differentiation. Another pro¬ cedure will give ajr, where a denotes the mean distance and r the radius vector, more directly, from the relation (f> — p -f e sin <£>. For differentiation with respect to p gives cld> I I + e cos 6 a " =-=-— = - = I + cos rp, dp I — e cos c p I — c- r suppose, when expressed in a Fourier series, and then 2 fir d .K 2 f 7T B,. = - I cos rp y dp. = ~ / cos r{tp - e sin <p)d<p = 2 j r (re), TTJ 0 dp TT J (I according to Bessel’s definition. An integration now gives f tre) <p = p + 22 J ——' sin rp r and 0 ~ p „I Are) . o sin <f> = - = 23 — sm rp ; &c. e re Chapters ii.-ix. are devoted to the purely analytical development of the Bessel function, considered as the solution of a differential equation, as an algebraical 01 trigonometrical series, or as a definite integral; these are the earlier chapters for which the authors apologise in the preface as appearing to contain a needless amount of tedious analysis. In Prof. Byerly’s treatise the re¬ quisite analysis is introduced in small doses, and only as required ; but the ordinary mathematician loves to strew the path at the outside with difficulties best kept out of sight ; thus, as Heaviside remarks, the too rigorous mathematician tends to become obstructive. It is of © 1895 Nature Publishing Group October 3, 1895] NA TURE 543 course reassuring to know that the functions employed in the physical applications, rest on a sound analytical basis, and that the convergency of the series has been carefully examined. But there is no compulsion to follow these demonstrations, tedious to all but pure mathema¬ ticians ; so we can pass on direct to Chapter x., where the physical interest is resumed, under the head of “ Vibra¬ tions of Membranes,” for instance the notes produced on a circular drum-head. Lord Kelvin’s oscillations of a columnar vortex, Lord Rayleigh’s waves in a circular tank, and Sir George Stokes’s investigation of the drag of the air in pendulum vibrations, make up an interest¬ ing Chapter xi. on Hydrodynamics. Chapter xii. deals with the steady flow of electricity or of heat, and Chapter xiii. with the fascinating and novel phenomenon of Hertz’s electromagnetic waves, when propagated along wires, in which problem the Bessel function assumes an essential importance. The Diffraction of Light, considered in Chapter xiv., contains important applications of the Bessel functions ; the hydrodynamical analogue would be the investigation of the effect of a breakwater in smoothing the waves which bend round behind into its shelter ; for instance, the effect of the Goodwin Sands on the safe anchorage in the Downs. Newton rejected the Undulatory Theory of Light, partly because he could not understand the existence of shadows on this hypothesis, a curious effect of Newton’s early ideas as a country boy ; had he been brought up on the sea coast, this apparent difficulty could not have troubled him. It would be a needless complication to consider any but straight waves in the case of the breakwater ; and similarly in the Diffraction problem, the authors might have made a simplification by parallelising the incident light by passing it through a lens ; or at least this special case, which is the one of practical importance in the subsequent discussion of the resolving power of a tele¬ scope, might receive separate treatment as the analysis now becomes almost self-evident. This chapter concludes with a discussion of Fresnel’s integrals, required in the diffraction through a narrow slit ; the integrals are ex¬ pressed by a series of Bessel Functions of fractional order, half an odd integer, and are represented graphi¬ cally by Cornu’s spirals. The problem of the stability of a vertical mast or tree, considered under the head of Miscellaneous Application in the last chapter, may well be amplified by examining the effect of centrifugal whirling on the stability, as in the case of the chain on p. I ; for the number of revolutions required to start instability is exactly equal to the number of vibrations which the mast or tree will make when swaying from side to side. A differential equation of the fourth order, with a variable coefficient, now makes its appearance, the solution of which will express the oscilla¬ tions of the bullrushes in a stream, or the waving of corn¬ stalks in a field. The curious appearance of permanence in the waves on a cornfield gives an illustration, analogous to Prof. Osborne Reynolds’s disconnected pendulum, of a case of zero group-velocity ; and by some intuitive deduc¬ tions from the appearance of these waves the farmer can judge the time suitable for harvest. The authors have been fortunate in securing an original NO. I353, VOL. 52] collection of numerical tables, including those of Dr Meissel, who did not live quite long enough to see his valuable calculations published in this book. A collection of examples adds greatly to the interest of the treatise, and will probably form the nucleus of a still larger list in the future. Altogether the authors are to be congratulated in bring¬ ing their task to such a successful conclusion; and they deserve the gratitude of the mathematical and physical student for their lucid and interesting mode of pre¬ sentment. A. G. Grrenhill. OUR BOOK SHELF. Protoplasms et Noyau. Par J. Perez, Professeur a la Faculte des Sciences de Bordeaux. (Bordeaux ; Imprimerie G. Gounouilhou, 1894.) EXPERIMENTAL work in recent years has repeatedly shown that in plants as well as in animals the physio¬ logical role of the nucleus in the cell is one of great importance. It has been demonstrated that non-nucleated fragments of protoplasm, whether of a Spirogyra or an Infusorian, are incapable of growth and reproduction ; while, on the other hand, fragments containing a portion of nuclear material are capable of complete recrescence. Impressed by these facts the writer of the essay before us has been led to doubt whether protoplasm can be properly regarded as the “ physical basis of life,” since it cannot retain its life when removed from the influence of the nucleus. Consistently with this position the writer throws doubt upon the existence of non-nucleate organ¬ isms in general. The presence of nuclei has been demonstrated in many forms once believed to be destitute of them— eg. Mushrooms, marine Rhizopods, and plas- modia. There remains only Haeckel’s group of Monera in which the presence of a nucleus may still be disputed. M. Perez considers in turn each of Haeckel’s subdivisions of this most artificial group. In the Lobomonera (eg. Protamceba ) he believes that the nucleus has been over¬ looked. In the Rhizomonera the nucleus has been observed in various species of Vampyrella; and it probably exists also in Protomyxa , since this form produces zoospores ; the zoospores of those Myxomycetes which most resemble Protomyxa have been shown by Zopf to be nucleated. In the Tachymonera (Schizomycetes) the greater part of the body seems to consist of nucleoplasm, while the zoogloea may perhaps be compared with the undivided protoplasm of a plasmodium. M. Perez concludes that non-nucleated organisms or cytodes are creations of the imagination ; that protoplasm, by which our author means cytoplasm, is not the primitive living matter, but a product of nucleoplasm ; and that nucleoplasm, and not protoplasm, is the most primitive living substance known to us. Analytical Key to the Natural Orders of Flowering Plants. By Franz Thonner. Small 8vo. pp. 151. (London : Swan Sonnenschein and Co., 1895.) The author’s apology for his little book is that few “Exotic Floras” contain artificial keys to the natural orders, even such as contain keys to the genera and species. But we imagine few persons would attempt working with a flora, exotic or native, without some pre¬ liminary knowledge of botany, and especially of the natural orders. Indeed a considerable acquaintance with the subject would be necessary to enable a person to use the present key to advantage. For example, the author begins with “ ovules naked,” and “ ovules enclosed in an ovary,” &c. Now, to be able to decide this point means a great deal, for a person who could do it would most likely know his gym nosperm without looking at the ovule © 1895 Nature Publishing Group