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volume. The subject is brought into touch with other branches of the science. Thus, under hydrocarbons, we read a little about thermochemistry; under aldehydes there is a reference to autoxidation; under acids there are a few words about steric hindrance; under ethereal salts (a rather antiquated term) a short account is given of mass action, and so forth. In addition to this there are separate chapters on laboratory methods, stereochemistry, the sugars, dynamic isomerism, heterocyclic compounds, and the physiological properties of organic compounds.

That the subjects are treated rather broadly than deeply seems no serious defect. They are sufficient for the general reader, who is provided with elaborate references if he desires to extend his knowledge. In conclusion, we would direct the author's attention to a few inaccuracies which have been noticed, and which might be modified or corrected in a future reprint. The two isomeric dimethylethylenes, which are stated to be known in only one form, have been prepared by J. Wislicenus (p. 310); the molecular weight of triphenylmethyl has been determined, and corresponds to the double formula (p. 423); Fischer and Slimmer were unsuccessful in effecting an asymmetric synthesis (p. 301); it is incorrect to state that propylene and hydrobromic acid give exclusively isopropyl bromide (p. 45).

We would also suggest the following:-Thiele's hypothesis requires amplification to be understood (p. 46); it is very questionable if the explosiveness of a compound depends upon its breaking up into stable molecules, for many silver salts share with silver oxalate this property, whereas a substance like platinic chloride does not explode; the statement that ethyl and methyl "cannot exist in the free state because they contain one of the carbon affinities unsaturated " (p. 27) is inconclusive, especially as triphenylmethyl is referred to later as possibly existing (pp. 36, 423); without some qualification it is misleading to say that Dumas's theory of types "was especially developed by Gerhardt," and "received the support of Williamson and Wurtz" (p. 17). In the first place, Williamson originated the idea of Gerhardt's types, which were simple inorganic compounds in which hydrogen could be replaced by radicals. They were intended to denote chemical behaviour and not relationships. Ether had no generic relationship to acetic anhydride, though they belonged to the same type. Dumas's types, on the other hand, were organic substances which were intended to show relationships produced by by substitution rather than chemical behaviour.

determination of physical constants such as are playing an increasingly important role in the study of structure. The descriptions are clear and full, and the photographic illustrations are masterpieces of their kind. Altogether the book is probably the most complete among those of home manufacture on the subject of practical organic chemistry that has yet appeared.

(3) This modest little volume, which is one of the University Tutorial Series, should form an excellent introduction to the study of organic chemistry, and if the process of practical instruction can be carried on concurrently with theoretical teaching, as the author does with his own class, nothing better could be desired. He takes a few of the commonest organic substances and uses them, as they can easily be used, to illustrate quite a large variety of chemical operations and products. If the substance of the book can be assimilated in the course of four school terms, as the author states, we think that both teacher and student should be satisfied with the result. May we suggest that the name of Wurtz should be spelt without the diæresis and Senderens without an a?

(4) One turns from the intricacies of a modern treatise on organic chemistry to Wurtz's classical memoir on the glycols with the same sense of relief that one listens to the simple melody of an early composer after the confused sounds of a modern orchestral symphony. Short and simple though it is, it is difficult to overrate the far-reaching results of Wurtz's research. It was not merely the discovery of a new class of alcohols and organic oxides, or an extension of Williamson's water type. It afforded for the first time clear experimental evidence of the existence of what were then termed "polyatomic " radicals. To quote Wurtz's own words :

"The main result, which, in my opinion, is derived from these synthetic experiments, is not the discovery of the new compound, glycol-there are enough new compounds in organic chemistry--it is not even the synthesis of glycerine nor the difficulties connected with its preparation which have been successfully overcome; but it is the manner of the formation of glycol and the antecedent reactions which made it possible; it is the conversion of the allyl compound by which the iodide passed into glycerine. All these experiments, which were directed to the same end, have shown that an organic group united to 2 atoms of chlorine or bromine can replace two atoms of silver, and are therefore equivalent to two atoms of hydrogen, and that an organic group united to three atoms of chlorine or bromine can replace three atoms of silver and is equivalent to three atoms of hydrogen."

The theory of polyatomic radicals, like ethylene and

elements or the theory of valency, upon which the whole fabric of modern organic chemistry rests. Wurtz himself held perfectly clear views on the different valency of the elements. In his address to the Chemical Society in London in 1862 on ethylene oxide, he points out that as ethylene oxide represents a diatomic radical united to oxygen, so many of the metals may be regarded as diatomic elements. The paper is well worth re-reading, and is not by any means the least interesting addition to the Klassiker. J. B. C.

(2) The "Practical Organic Chemistry" of Sud-glyceryl, soon developed into that of the polyatomic borough and James is rather a laboratory handbook or book of reference than a course of practical instruction. As stated in the preface, examples are given of different types of operations. These types are grouped together. Thus, there is a chapter on the preparation of hydrocarbons, one on alcohols, another on halogen compounds, acids, esters, nitro-compounds, sulphonic acids, and so forth. In addition, there is a preliminary chapter on organic analysis and molecularweight determinations, and, at the end of the volume, a number of useful examples of analyses and the


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(1) The Elements of Electricity and Magnetism. A
Text-book for Colleges and Technical Schools. By
Prof. W. S. Franklin and Barry Macnutt.
The Macmillan
viii+351. (New York:
London: Macmillan and Co., Ltd., 1908.) Price
75. net.

(2) A Short University Course in Electricity, Sound
and Light. By Dr. Robert A. Millikan and J.
Mills. Pp. v+389. (Boston and London: Ginn
and Co., n.d.) Price 8s. 6d.

(3) Naturlehre für höhere Lehranstalten auf Schuler-
übungen gegrundet. By Dr. Friedrich Danneman.
Teil ii. Physik. Pp. vii+204. (Hanover and
Leipzig: Hahnsche Buchhandlung, 1908.) Price
3.50 marks.

(4) The Elementary Theory of Direct Current Dynamo
Electric Machinery. By C. E. Ashford and E. W. E.
Kempson. Pp. vii+120. (Cambridge: University
Press, 1908.) Price 3s. net.

(5) Electrical Laboratory Course for Junior Students.
By R. D. Archibald and R. Rankin. Pp. vi+95.
(London: Blackie and Son, Ltd., 1908.) Price

Is. 6d. net.

(1) THE order in which the elements of electricity and
magnetism are presented in the first volume
under notice is :-(a) Electric current; (b) magnetism;
This order is one
(c) electrostatics; (d) electric waves.
which does not make the exposition perfectly happy.


Thus it does not seem natural when it is found neces-
sary to refer provisionally to the measurement of cur-
rents by their magnetic effect (p. 7) prior to any state-
ment as to how magnetic effects themselves
measured. Surely the natural order is to take mag-
netism before considering the electric current, even
though it may be preferred to deal with both before
considering the phenomena of electrostatics.
author passes naturally and easily to the exposition
of the last-named phenomena, and as many prefer
this order this portion may certainly be commended to


The author is convinced that "elementary science instruction must be made to touch upon the things of everyday life if it is to be effective." This sentence may be taken as the keynote to the entire book. Thus electric resistances are usually represented as electric lamps. Those who are accustomed to abstract thinking may smile at these concrete representations; but it must be remembered that this is only an elementary book, and it must be admitted that much of the difficulty which many junior students feel is connected with the unreality of the subject as it appears to them. We commend the book for endeavouring, in this and other ways, to make the subject more real than it usually is.

More attention is given than is customary in an elementary course to phenomena connected with recent discoveries, such as kathode rays, radio-activity, electric waves, &c. We conclude that in America a junior course is in some respects more advanced than with


This remark applies most to the chapter on elec

tric waves, and to the appendix on ship's magnetism.
Both these portions are very well done, though we
would have thought them fairly strong meat for those
who are making a "first systematic study of the
subject." However, whether a student takes
in completely in his first study or not, he will be very
glad to find them here ready to hand when required.
(2) The second of the volumes under review" repre-
sents primarily an attempt to secure a satisfactory
articulation of the laboratory and class-room phases
of instruction in physics." Expressed otherwise, it
consists of a description of laboratory work, each
experiment being preceded by as much theory as is
necessary to make a complete logical exposition of the
subject under study. We think that this plan is an
excellent one; and it has been very satisfactorily
carried out. Of course, it will be understood that the
theoretical part is not sufficient to replace a text-book
dealing specially with the theory.

Although electrostatics is introduced in the first chapter, electric capacity is not defined until later, when it can be measured by means of a ballistic galvanometer. There may be something to be said for this, but we think that the course of experiments would be considerably improved and the student would get a more vivid idea of what capacity is if experi

ments were added on parallel plate condensers used along with a gold-leaf electroscope or an electrostatic voltmeter.

In sound, a series of experiments on diffraction is included. The experiments on light begin with diffraction, a fact which prepares us for the exclusive use of the wave-method in proving the general phenomena of reflection and refraction. The final chapter is on radio-activity, and contains some simple experiments on uranium and thorium salts. The book is altogether a most excellent manual.

(3) We find in our third volume a well-selected series of very elementary experiments in the whole round of physics suitable particularly for school use. Though the subject is dealt with satisfactorily as a rule, it is not beyond criticism. The diagram of the paths of rays in a microscope would be improved if the rays represented as passing through the eyepiece were the "' would be same as those transmitted through the objective. The experiment on the "velocity of electricity best left out of such an elementary book; the statement that electricity travels with the velocity of light is, of course, absurd.

(4) We are in entire agreement with the authors of this book, that in the training of an electrical engineer there should be included a knowledge of the theory of the subject built up logically from first principles, each step being illustrated with the help of some piece of machinery or practical appliance of a general and simple rather than an elaborate or necessarily up-todate type. The present volume is intended to be used only as a note-book accompanying a course of experimental lectures. The authors are to be congratulated on the excellence of their little manual. The diagrams in particular are very carefully designed.

(5) The last of this group of text-books covers an elementary first year's evening course and part of a

second year's course. The first method of proving the inverse square law for magnetic poles will not convince. However, putting aside an occasional criticism of this kind, we think that the book will well serve its purpose of replacing manuscript instruction sheets in a junior laboratory.

OUR BOOK SHELF. Théorie des Corps déformables. By E. et F. Cosserat. Pp. vi+226. (Paris: A. Hermann et Fils, 1909.)

Price 6 francs.

THE authors, who are well known by their writings on general elastic theory, here reprint in separate form an appendix contributed by them to M. Chwolson's Traité de Physique. The kinematical and dynamical theories of the flexible line, the flexible surface, and the deformable three-dimensional medium

are discussed in turn in great detail. The dynamical standpoint adopted is that of the principle of action, which forms, in the authors' opinion, the only satisfactory basis for the "deductive" exposition of the subject. In each case the most general form of the function representing the "action" is sought which is consistent with the necessary invariantive relations. This procedure is, of course, not altogether new, and an expert, turning over the pages, will recognise much that in one form or another is familiar to him. The treatment is necessarily somewhat abstract, and is mathematically very elaborate, Cartesian methods being followed throughout. To many readers the long train of general investigations, unrelieved by a single application, may prove deterrent; but the authors at all events claim that their procedure has never before been carried out so resolutely and completely, and may justly pride themselves on the mathematical elegance of their work. Apart from its other qualities, the treatise has a distinct value as a book of reference, and furnishes a whole arsenal of formulæ which may save trouble to future writers.

The book begins with a kind of philosophical introduction to which the authors attach great importance. This requires to be read in conjunction with a previous treatise, which has also appeared in the French edition of M. Chwolson's work. Those who adopt in its fullest extent the empirical view of mechanics will perhaps consider that too much weight is attached to discussions of this kind. The historical references are, however, interesting, and fairly complete. The authors are indeed exceptionally well read in the history of their subject, and admirably conscientious in their citation of authorities. In their preface they promise a subsequent treatment of the theories of heat and electricity from a similar standpoint.

Practical Physiological Chemistry. A Book designed for Use in Courses in Practical Physiological Chemistry in Schools of Medicine and of Science. By Prof. Philip B. Hawk. Second edition, revised and enlarged. Pp. xvi +447. (London J. and A. Churchill, 1909.) Price 16s. net.

PROF. HAWK's text-book falls into the front rank with the numerous additions and improvements which have been introduced into the new edition. It is not only a practical guide, and, as such, should be found in all physiological laboratories, but forms a very complete, readable, and up-to-date account of our present knowledge of the chemical side of physiology. A special feature has been made of the illustrations, which are beautifully executed, and most of which will be new to workers in physiological chemistry. The crystalline forms of the many protein derivatives which the work of Emil Fischer and his colleagues

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has been instrumental in rendering familiar to the students of this branch of science will be found among them.

One small slip we notice in connection with the matter of protein nomenclature. The initiation of the new system of terminology which is now being adopted for the albuminous substances is wrongly attributed to the British Medical Association. It was really a committee of the Physiological and Chemical Societies of this country which set the ball rolling.

The mistake is, however, a pardonable one, seeing that it was at the meeting of the British Medical Association held at Toronto in 1906 that the opportunity of presenting the subject to our American colleagues was taken advantage of. The success that has attended this effort to secure uniformity of nomenclature amongst English-speaking people has been very gratifying; the American system, adopted under the auspices of the American Physiological Society differs in only small and unimportant details from and the American Society of Biological Chemists,

our own.

W. D. H.

Behind the Veil in Bird-land. Some Nature Secrets revealed by Pen and Camera. By Oliver G. Pike, with a number of pen sketches by E. R. Paton. Pp. 106. (London: The Religious Tract Society, 1908.) Price 10s. 6d. net.

SINCE the Keartons, some years ago, showed what splendid results could be achieved by an intelligent use of the camera as an aid to the study of natural history, a host of nature-photographers has arisen, but only a very few have attained the high standard of merit set by the founders of this branch of photography. Mr. R. B. Lodge and Miss E. L. Turner in this country, Schillings in Germany, and H. K. Job in America have in some respects even surpassed the Keartons; while in this display of resource and dogged persistence in the most trying circumstances they stand unrivalled.

Mr. Pike in this rather pretentious volume has given some very excellent_photographs, but the "Nature Secrets revealed by Pen and Camera" which he promises in his title-page are conspicuous by their absence. His pages contain hardly one single new fact, but a great deal that is banal. He solemnly assures us, in writing of the kestrel, that "The first summer rose, a delicate pink amidst the surrounding green, is a greater picture of spring than ever the sunlit sea could be "--which statement contains a great deal of truth!" and," he continues, "a kestrel hovering over a meadow, yellow with summer's flowers, tells us a deeper story than the eagle soaring over a wind-swept moor." We fail to grasp why this should be so.


Bird-land's veil" is constantly being "lifted up " for him, like the drop-scene at the theatre, and on the stage appear blackbirds, which tell him" the story of the leaves and flowers," and wrens, which reveal "the secrets of the hedgerows," while skylarks, to complete the illusion, like the celebrated Grigolati troupe in the pantomime, fly to and fro across the stage, and sing "happy songs"! Perfectly charming! W. P. P.

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An Account of the Deep-sea Asteroidea collected by the R.I.M.S.S. Investigator." By Prof. René Koehler. Pp. 143; 13 plates. (Calcutta: Indian Museum, 1909.) Price 12 rupees.

THIS substantial contribution to the material of the echinoderm "system" consists of 126 pages of minute description, and nine pages of general remarks. It is a continuation of certain reports of a preliminary and incentive character published many years ago by the naturalists and pioneers of the Indian Marine Survey, but, except that some doubtful identifications

are disposed of and some errors criticised, it does not incorporate that earlier work.

In the descriptive part of the memoir thirty-nine species are enumerated, of which thirty are regarded as new, and are exhaustively described. The general remarks refer to eighty-eight species-the thirty-nine species treated by the author, and forty-nine species dealt with in the earlier reports and furnish the evidence of the author's main conclusions. These conclusions are that the deep-sea starfish of the Bay of Bengal and Arabian Sea are much more Phanerozonia than Cryptozonia, and that their geographical affinities, so far as they can be discerned at all, are exclusively Indo-Pacific, with a slight Hawaiian touch.

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tensile, shock, bending, hardness, and torsion tests.
Chapter i. shows the influence of annealing and of
cold work. Chapter iii. is devoted to "étirage," or
drawing, defined as an operation which has for its
object the completing of work done by rolling and
giving to the metal a cross-section which cannot be
obtained by rolling," after the manner of wire-drawing
("tréfilage," chapter iv.), which is a special case of
drawing where the cross-section is circular. Chapter v.
gives a short account of methods of straightening
(" dressage ").
A. McW.

Nutrition and Evolution. By Hermann Reinheimer.
Pp. xii+.284. (London: John M. Watkins, 1909.)
Price 6s. net.

THIS is an essay on the importance of nutrition as a
factor in evolution, and the author is in good com-
pany. For was it not Claude Bernard who said,

Of the new species described by Prof. Koehler, five are separated as types of new genera. These are Johannaster, which is placed with very justifiable hesitation among the Plutonasteridæ, for some of its characters suggest a pentagonasterid connection; Phidi-"l'évolution, c'est l'ensemble constant de ces alternaaster, which seems scarcely distinct from Psilaster; Sidonaster, which agrees in all points with Porcellanaster, except that, as in other porcellanasterid genera, the elements of the cribriform organs are papillar instead of lamellar; and Circeaster and Lydiaster, both of which are Antheneids having the abactinal plates of the disk much smaller than those of the rays.

tives de la nutrition; c'est la nutrition considerée dans sa réalité, embrassée d'un coup d'œil à travers le temps"? To have had this thesis worked out in a methodical manner would have been great gain, but the author is not strong in scientific method. He has gleaned far and wide to illustrate "the evolutionary aspects of nutrition," and while he has a crow to pick with most of his authorities, who have not the central key of a uniform analysis," he uses them when they suit him to back up his conclusion "that in its silent effects nutrition is one of the most formidable factors in the shaping of individual and racial destinies." The conclusion is sound, but we cannot say this of many of the arguments.

It may be thought that the limits of some at least" of these genera are cut too fine to last; and of the descriptions of species it may almost be said that they are accurate expositions of specimens rather than impressive definitions of nature's products; but such is the way of systematic zoology nowadays.

The memoir is most bountifully and most beautifully illustrated by the author's own hand; the plates, which are thirteen in number, are quite above criticism. Antimony its History, Chemistry, Mineralogy, Geology, Metallurgy, Uses, Preparations, Analysis, Production, and Valuation; with complete Bibliographies for Students, Manufacturers, and Users of Antimony. By Chung Yu Wang. Pp. x+217; illustrated. (London: C. Griffin and Co., Ltd., 1909.) Price 12s. 6d. net.

MR. WANG observes in his preface that a metallurgical work in English by a Chinese author is unusual. After reading the book, the conclusion is irresistible that English metallurgists would gain if Chinese authors were more numerous. Mr. Wang has treated his subject with the greatest respect, and has drawn up with methodical care a complete treatise which will be very useful to all students of the subject. The long and apparently exhaustive bibliography at the end of each chapter would alone give the book a right to a place on metallurgists' shelves, but in many cases the facts are sufficiently set forth in the present work.

The author carried out some practical tests of the latest volatilisation process of extracting antimony from its ores, which was patented last year by M. Herrenschmidt, and seems to have been much impressed by its merits. The account of these tests is, however, almost the only original matter in the book, which is mainly a compilation of previously published material, printed without comment. Its merits lie chiefly in the logical sequence and the accuracy of the extracts. Etirage, Tréfilage, Dressage des Produits métallurgiques. By M. Georges Soliman. Pp. 164. (Paris: Gauthier-Villars and Masson et Cie., n.d.) Price 3 francs.

THIS interesting little work, one of the well-known "Aide-Mémoire" series, deals with its subject from a practical point of view. It is divided into five chapters, the first considering shortly the general mechanical properties of metals and alloys such as


[The Editor does not hold himself responsible for opinions expressed by his correspondents. Neither can he undertake to return, or to correspond with the writers of, rejected manuscripts intended for this or any other part of NATURE. No notice is taken of anonymous communications.]

Bessel's Functions.

I ONCE stated that a good style of writing English is not a strong point amongst British mathematicians, and the justice of this remark is exemplified by Prof. Hill's letter on this subject (NATURE, July 8), since it contains the phrases Meissel's tables, Smith's tables, Aldis' tables, Isherwood's tables, which are correct; and Bessel functions, British Association tables, which are wrong. It is not in general permissible in English to employ a proper noun as an adjective, for the rules of grammar require either the use of the genitive case, or the conversion of the noun into an adjective, as in the words Newtonian, Lagrangean.

The British Association is one of the most important societies in the British Empire; it long ago discarded the insularity of our ancestors, and has become cosmopolitan in its operations. It is therefore not too much to expect that it will conform to the rules of grammar in its publications, and employ its influence in encouraging a good literary style.

I do not understand what Prof. Hill means by Neumann's functions. I believe that Neumann was the first mathematician who studied the properties of zonal harmonics and allied functions of degree n+1, where n is zero or a positive integer; but the subject was afterwards taken up and greatly extended by Prof. W. M. Hicks in connection with circular vortex motion. Hicks calls these harmonics toroidal functions, which is a much better phrase, since it puts in evidence the fact that these functions are connected with the potentials of anchor rings or tores.

There is also another class of functions which are zonal harmonics of complex degree in-. These have been studied by Hobson (Trans. Camb. Phil. Soc., vol. xiv., p. 211), who calls them conal harmonics.


Fledborough Hall, Holyport, Berks, July 9.

Musical Sands.

MAY I record the existence of musical sands along the shore at the Sandbanks, Poole Harbour?


Some years ago the Poole authorities erected a series of box groynes along this coast between Poole Head and the Haven, and these have considerably increased the natural accumulations of sand, so that it is making everywhere, and the growth of the marram grass on the dunes is in many places (independently of that recently planted) rapidly extending seawards.

The beach now, between each groyne, consists of wide and flat deposits of sand, shells, and flint pebbles, but about midway between the dunes and the sea, where the sand is comparatively free from these, musical zones are of frequent occurrence.

In walking along the shore in a westerly direction, starting from the first groyne, the sounding qualities of the sand notably increase. Thus between the first and second groynes there are no musical patches, between the second and third the sounds are very faint, and between each of the other groynes, until one reaches the last at the Haven Point, the intensity of the sound increases. In a small cove at the Point, formed by the last groyne (constructed of barrels of concrete and an old ship), the sand is remarkably musical.

The increase of sound observed when walking in a westerly direction is due, I think, to the fact that the prevailing westerly winds, and the littoral drift, separate the finer particles from the sand and carry them eastwards, and a microscopic examination of samples obtained from distances about a mile apart on this shore confirms this. This musical sand is of the Studland Bay type, and near the Haven gives even better results than any I have found there. The occurrence of musical sands along this particular shore through the conserving influence of the groynes is an interesting fact, for their existence there previously was very unusual, being only once noted in very small quantity during the last twenty years. Parkstone-on-Sea, July 4.



The Commutative Law of Addition, and Infinity. REFERRING to the review of Hilbert's Grundlagen der Geometrie, on p. 394 of No. 2066 of NATURE (June 3), may I point out that the commutative law of addition can be proved without the help of any axioms at all, other than those of general logic? The method, indeed, used by Peano in 1889 (Arithmetices Principia .," Turin, 1889, p. 4), which is only based on axioms of a general nature (such as the principle of mathematical induction), and not on such special laws as the distributive ones, appears in so far superior to Hilbert's; and, since all Peano's axioms were proved in Mr. Russell's "Principles of Mathematics " of 1903, Hilbert's proof seems quite superseded. Further, the difficulties arising out of Dedekind's proof of the existence of infinite systems can be avoided without the introduction of "metaphysical arguments about time and consciousness (see Russell, Hibbert Journal, July, 1904, pp. 809-12), as, indeed, your reviewer seems to think possible. But the connection of the fact that the existence of an infinity of thoughts (which must be in time) with Hamilton's idea that algebra was interpretable especially in the time-manifold, just as geometry is in the spacemanifold, is not obvious. PHILIP E. B. JOURDAIN. The Manor House, Broadwindsor, Beaminster, Dorset, July 2.

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NEITHER Dr. Hilbert nor the reviewer make any suggestion that the commutative law of addition is best proved as a deduction from the laws of multiplication. But the laws of multiplication are so often treated as deductions from those of addition that it is interesting to have a case of the converse procedure. The fact that both these operations and their laws have been treated independently and in a strictly logical manner by Dedekind, Peano, and others is, of course, perfectly well known to all who have paid any attention to this part of mathematics. Whether Dedekind's critics have really avoided metaphysical arguments without at the same time making metaphysical assumptions is a question on which a difference of opinion is permissible. G. B. M.

THE THEORY OF CROOKES'S RADIOMETER. I HAVE noticed that the theory of this instrument is usually shirked in elementary books, even the best of them confining themselves to an account, and not attempting an explanation. Indeed, if it were necessary to follow Maxwell's and O. Reynolds's calculations, such restraint could easily be understood. In their mathematical work the authors named start from the case of ordinary gas in complete temperature equilibrium, and endeavour to determine the first effects of a small departure from that condition. So far as regards the internal condition of the gas, their efforts may be considered to be, in the main, successful, although (I believe) discrepancies are still outstanding. When they come to include the influence of solid bodies which communicate heat to the gas and the reaction of the gas upon the solids, the difficulties thicken. A critical examination of these memoirs, and a re-discussion of the whole question, would be a useful piece of work, and one that may be commended to our younger mathematical physicists.

Another way of approaching the problem is to select the case at the opposite extreme, regarding the gas as so attenuated as to lie entirely outside the field of the ordinary gaseous laws. Some suggestions tending in this direction are to be found in O. Reynolds's memoir, but the idea does not appear to have been consistently followed out. It is true that in making this supposition we may be transcending the conditions of experiment, but the object is to propose the problem in its simplest form, and thus to obtain an easy and unambiguous solution-such as may suffice for the physicist will naturally wish to go further. We purposes of elementary exposition, although the suppose, then, that the gas is so rare that the mutual encounters of the molecules in their passage from the vanes to the envelope, or from one part of the envelope to another part, may be neglected, and, further, that the vanes are so small that a molecule, after impact with a vane, will strike the envelope a large number of times before hitting the vane again.

Under ordinary conditions, if the vanes and the envelope be all at one temperature, the included gas will tend to assume the same temperature, and when equilibrium is attained the forces of bombardment on the front and back faces of a vane balance one another. If, as we suppose, the gas is very rare, the idea of temperature does not fully apply, but at any rate the gas tends to a definite condition which includes the balance of the forces of bombardment. the temperature be raised throughout, the velocities of the molecules are increased, but the balance, of course, persists. The question we have to consider is what happens when one vane only, or, rather, one face of one vane, acquires a raised temperature.


The molecules arriving at the heated face have, at any rate in the first instance, the frequencies and the velocities appropriate to the original temperature. As the result of the collision, the velocities are increased. We cannot say that they are increased to the values appropriate to the raised temperature of the surface probably require numerous collisions. Any general from which they rebound. To effect this fully would increase in the velocity of rebound is sufficient to cause an unbalanced force tending to drive the heated surface back, as O. Reynolds first indicated. If we follow the course of the molecules after collision with the heated surface, we see that, in accordance with our suppositions, they will return by repeated collisions with the envelope to the original lower scale of velocities before there is any question of another collision with the heated face. On the whole, then,

1 See for example Poynting and Thomson's "Heat," p. 150.

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