54-^ 



NA TURE 



[October 3, 1895 



erosion and denudation have modified the primar>' 

 features on a gigantic scale ; 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 

 deterjnined 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. W'e sincerely 

 desire, in the interests of geography, that at least these 

 new volumes of the scries should be rendered accessible 

 to West European geographers. 



The described region is verj- thinly populated, and 

 contains but few explored remains of the past. ."Vs 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 summar)- of all that is known about 

 the fauna of the lake. P. K. 



APPLICATIONS OF BESS EL FUNCTIONS. 

 A Treatise on Ressel Functions and their Applications to 

 Physics. By Andrew tira>', M.A., and G. B. Mathews, 

 M..\. (London : Macmillan and Co., 1895.) 



THIS 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. TodhunteHs " 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. I, 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 

 \ertical 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 extraordinar)- 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 

 Ittcnty-fivc years, and we find Boole's " iJiffercntial 

 Kqualions " ignoring the Bessel Function and the solution 

 of the general Kiccation ecjuation which it provides. In 

 those days it was ruslomary 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 hybcrbolic functions were tabooed. 



Siring is the fa\ourite material of the mathematician 

 for illustrating catenary propLMties ; but it is a relief to 

 find that the authors ha\e pro\ided 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. i), 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 kinkiness, for the representation 

 of catenaries and their oscillations, is at once manifest. 



The small plane oscillations of the chain about its 

 mean vertical position arc 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„ ( ^.r) 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, I '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 

 ii = nt ■\- ( - ra. 



The authors avoid the awkward integration by parts 

 emploNcd by Todhuntcr in determining the excentric 

 anomaly <(> by means of a dift'creniiation. Another pro- 

 cedure will give air, where a denotes the mean distance 

 and r the radius vector, more directly, from the relation 



^ = ^ + csin^. 

 For difierentiation with respect to /u gives 



d^ _ I _ I + e cos 6 



dn I - (T cos ^ I - 



= - = I + HUr cos rfi. 



suppose, when expressed in a Fourier scries, and then 



B,. = ? ( ' cos r/i^^d/i = - I cos r(0 - ^siii tpycp = 2T,-f><). 

 irj dfi ir f I* 



according to Bessel's definition. 

 .\n integration now gives 



and 



sin 



ip = H + 2Si^^^ sin r/i 



= ^ '^ = 2a sm r/jL ; &c. 



Chapters ii.-ix. arc devoted to the purely analytical 

 development of the Bessel function, considered as the 

 solution of a differential equation, as an algebraical or 

 trigonometiical 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 Hcaviside remarks, the too rigorous 

 mathematician tends to become obstructive. It is of 



