Skip to main content

Full text of "A Treatise on Bessel Functions and their Applications to Physics"

See other formats


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