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great foods of the Seine; and thirty-one tables are has involved an error in work, which, however, is a appended at the end of the volume, giving the rain- mere slip. The theorem of mean value is very well fall, discharges, and water-levels at different dates in explained and used in the deduction of Taylor's various parts of the Seine basin, and eleven sets of theorem for the determination of the remainder, a graphic curves indicating the decrease in the dis- little geometrical figure assisting the student to undercharges of the Seine, some of its tributaries, and stand the nature of this remainder. (Correct, howcertain sources, at different periods. Table xxiii., ever, the errors in sign in the first equation of giving the rainfalls of the warm seasons, and the p. 169.) high floods of the following cold seasons, at the Auster- The discussion of the convergency and divergency litz Bridge, Paris, and at Mantes, from 1874 to 1900, of series is very good, and a somewhat uninteresting shows that none of these warm seasons in which the subject is rendered simple and attractive. An inrainfall was below the mean of 14.88 inches, was cautious statement, however, is made with regard to followed by floods of the Seine rising higher than an alternating series, p. 241, according to which if we 14.44 feet on the gauge at Paris, and 19.72 feet at stop at the nth term of such a series the error made Mantes; and the eight cold seasons in which the is numerically less than the value of the (n + 1)th Seine reached or exceeded 16.40 feet at Paris, and term. Clearly this is not in general true if the 21.06 feet at Mantes, were all preceded by warm alternating series is one in which the numerical values seasons in which the rainfall exceeded the mean. of the terms increase for a while and then diminish. Moreover, with the exception of 1890, when the warm For example, the series for sin x is an alternating season came between two very dry cold seasons, all one of this kind. If x=5, the numerical values do the warm seasons having a rainfall above the average not begin to diminish until after the third term. The have been followed by foods of the Seine, attaining property asserted, and the proof in p. 226, must be at least 10.17 feet at Paris and 16.40 feet at Mantes; applied to cases in which we stop after the greatest whereas none of the fourteen warm seasons with a numerical term has been passed. rainfall below the average was succeeded by floods The theory of maxima and minima is well illusin the next cold season, reaching the height attained trated by examples taken from various branches of in eight of the cold seasons preceded by warm seasons physics. Even at the risk of being a little hyperin which the rainfall exceeded the average,

critical, we must, however, point out that the time

taken by a ball to roll down a plane the base of A NEW AMERICAN WORK ON THE which is of length a and the inclination of which is CALCULUS.

$ is not 2 Na/g sin 2$, as it is said to be in p. 128, Elements of the Differential and Integral Calculus.

for the simple reason that the acceleration of the By William Anthony Granville, Ph.D., with the centre of the ball (if the ball is solid and homoeditorial cooperation of Percey F. Smith, Ph.D. geneous) is not g sin $, but 5/7 g sin $. This fact Pp. xiv + 463. (Boston and London : Ginn and Co.) is of importance in dynamics, and the matter should Price 1os. 6d.

be set right.

The part of the book dealing with curves is very THIS

is a book the main object of which seems to be to enable the student to acquire a knowledge systematic manner in which (pp. 267, 268) the student

good, and, in particular, we would commend the of the subject with little or no assistance from a

is taught to trace a curve from its equation. teacher; and, after a very careful study of it, we are

In the portion dealing with the integral calculus enabled to say that the work is admirably constructed

an exhaustive exposition of all the devices used in for the purpose. There is a complete absence of the

integrating functions is given. The reduction stilted formality which is usually supposed to be

formulas to be applied to the binomial integral appropriate to a mathematical treatise. In foot-notes, and sometimes in the text, the student is given scores { xm(a

xm(a + bx")pdx are given in tabular form on p. 345, of useful hints and warnings against errors into which and the student is told very properly that he should he would probably fall. Thus the work possesses a not memorise them. Instead of memorising them, very high value for the student; and it will be found he should apply a single simple rule which was given no less helpful to the teacher, for it contains a very long ago by Hymers in his “ Integral Calculus." large number of examples in every part of the subject, This rule enables us to obtain, without an effort of while it abounds in excellent diagrams.

memory, the exact formula appropriate to the reducThe portion on the differential calculus occupies 285 tion of any given binomial integral. pages, and terminates with 6 pages containing Besides areas and volumes (accompanied by excel. nothing but figures of all the curves more or less lent figures), polar moments of inertia of plane areafamous which present themselves in the subject, such are dealt with. The author speaks of these a. as the conchoid of Nicomedes, the cycloid, the moments of inertia about “a point "-an expression catenary, the cissoid of Diocles, the probability curve, which leaves something to be desired, since it in various spirals, &c.

always an axis that is involved. What we always The work is very strictly logical in its method—here require in this connection in dynamics is the mean and there a little too much so, perhaps.

square of distance of a body from an axis, and he Thus in p. 97 the proof that the angle between the should look to writers on the calculus to emphasise radius vector and the tangent to a curve has rdo dr this notion of a mean square of distance, instead of for its tangent is quite unnecessarily accurate, and the “square of the radius of gyration," k?. The

for dy.


student might easily learn to regard k as the distance practically as well as living cultures; and, though of mean square, just as we speak of the velocity of the increased time required to obtain a result and the mean square in a gas.

slight loss of delicacy render the use of living The book has a useful chapter on the simpler forms cultures still desirable in the laboratory, the safety and of differential equations, and concludes with a figure convenience of the dead cultures place the “ test and description of the integraph for finding the area within the personal practice of every not-too-busy of a curve. It might well include a description of practitioner. It is, however, strange to read (p. 13) Amsler's planimeter, and show how it finds areas, that the use of an oil-immersion objective is necessary. positions of centres of gravity, and moments of inertia The author then considers briefly the agglutination of plane figures; and, as to the proof of the theory phenomena found in tuberculosis, dysentery, and other of Amsler's planimeter, it need occupy no larger space diseases. Serum diagnosis of tubercle is considered to than the area of a shilling, notwithstanding the length be of very doubtful value. Appropriate stress is laid and complication of proofs which are usually given. on the fact that in many diseases (especially plague

The author's attention may be directed to the follow- and cholera) agglutination, in comparison with other ing misprints :-P. 44, note, Leibnitz was Gottfried, symptoms, is of very little use for the direct diagnosis not Gottfreid; p. 206 (A), read f's for fri p. 216, of the disease, though of the greatest value in the Ex. 15, read ,- for vo; P. 225, line 5, read 223 for 225; identification of the isolated organism. This part of p. 275, line 6, read Pl for P; p. 374, line 1, read

у the book is, however, less satisfactory than the earlier GEORGE M. MINCHIN. sections. Indeed, the serum diagnosis of Malta fever

is not mentioned, though the practical value of the

phenomenon in the diagnosis of this variable and often SERUM DIAGNOSIS.

very obscure disease has been demonstrated beyond. Manual of Serum Diagnosis. By O. Rostoski. question. Authorised translation by Charles Bolduan. Pp. The book concludes with an account of the identifivi +96. (New York: J. Wiley and Sons; London: cation of blood stains by the precipitin test. Readers Chapman and Hall, Ltd., 1904.) Price i dollar. will find here a wise injunction to make sure that any THIS THIS small work forms a companion volume to given stain is blood before deciding whether it is of

that by Wasserman “ Hæmolysins and human or animal origin; the precipitin will not disCytotoxins," which has already been noticed in these tinguish between the different tissues of the same columns. Each volume forms a monograph on some species of animal in the same way as it will separate part of those newer developments of bacteriology the same tissue from different species.. which concern immunity and kindred subjects. The In the translation several useful additions have been aim of the series is to provide simple yet compre-made; the last chapter, which attempts an impossibly hensive accounts of our present knowledge suitable precise and entirely arbitrary definition of the Widal for those who do not make a special study of the reaction, might, however, well have been omitted. Laboratory aspects of disease rather than exhaustive

A, E. B. treatises adapted for special students. That the information is authoritative and trustworthy is vouched

HISTORY OF PHARMACY. for by the list of authors, which includes some of the

Geschichte der Pharmazie. By Hermann Schelenz. most distinguished names in contemporary bacteri

Pp. ix +934 (Berlin: Julius Springer, 1904.) ology. Each volume is the work of one who has

Price 20 marks. himself made important contributions to the study of the subject.


HE successful practice of pharmacy implies some The present volume deals with the practical use of

acquaintance with plant chemistry and with agglutinins, bacteriolysins, and precipitins in diag- that branch of economic botany known as materia nosis. More than two-fifths of the whole is devoted

medica. For this reason the history of pharmacy, to an account of the Widal reaction in enteric fever. although it appeals particularly to the pharmacist and This section is extremely good, and for it alone the the physician, presents also many points of interest book is well worth reading. The author points out to the chemist and the botanist. Herr Schelenz does very clearly that the "test" is not to be regarded

not consider that the classes of readers here enumeris more than the “ first of the cardinal symptoms of ated form a sufficiently wide circle for his purpose, * phoid." Some discredit has been cast on the value and he states in the preface to this volume that he of the reaction, because clinicians have not always hopes also to interest the legislator, the antiquarian, found that infallibility which is so often expected of and the philologist. the laboratory, but which can never be present in

The book begins with a description of the condealing with so variable a complex as living matter. ditions under which pharmacy was practised among Removed from the pedestal of a “test” to the the Jews. A summary of the political history of the common ground of a “symptom,” the phenomenon nation is first given, and this is followed by sections tem, to have a better chance of receiving the appre- dealing with Biblical and Talmudic references to the ciation which it deserves. There is an admirable practice of pharmacy and the social condition, &c., account of the mixed and “group agglutinations of the practitioners of the art. The most interesting in typhoid and paratyphoid infections, and due notice portion of this section is that describing the drugs is taken of the use of typhoid cultures which have employed by Jewish apothecaries. It is curious that been killed by the addition of formalin. These react so many of these are still in use at the present day; for example, myrrh, Indian hemp, cassia (or cinna- those to any of the other sections. The text is, in mon), coriander, colocynth, galls, almonds, galbanum,

fact, a concise synopsis of the leading groups of birds, and storax are among those mentioned by the author.

with special reference to the specimens exhibited in

the galleries. The plan of the synopsis necessarily The Jews also appear to have made use to some ex

follows the system adopted in the museum, and it tent of natural mineral waters and various medicated

would therefore be quite out of place to criticise that baths as remedial agents.

system on the present occasion. A similar remark Similar accounts of the practice of pharmacy among applies to the fact of the illustrations (which are the Phænicians, Assyrians, Egyptians, Hindus,

admirable of their kind) being taken from the stuffed Persians, Greeks, Romans, and other peoples are

specimens in the collection instead of from living

birds--the guide is to illustrate the collection, and given, and then this racial method of treatment is therefore it is quite right and proper that the figures interrupted, and a chapter is inserted giving an of the birds should be taken from those shown in the account of the methods of the professors of magic, gallery. In addition to the general synopsis, there is astrology, and alchemy in the Middle Ages, and show- a guide to the series of British nesting birds. That ing how the practice of these secret arts gradually

the general plan and execution are in the main excel

lent cannot be denied; whether it will suit the taste led to a knowledge of natural science.

-and the purses of the public remains, however, to Resuming his narrative after this digression, the

be seen. author deals with pharmacy among the Copts and When a new edition is called for, certain emendaSyrians, the Arabs, and the Teutonic races, and tions may with advantage be made in the text. The brings it to the close of the eighteenth century with

most serious error we have detected is the statement

(p. 11) that the largest Æpyornis was probably not a short account of the condition of medicine and

more than 7 feet in height, whereas there are actually pharmacy in Italy, when the school of Salermo exer- limb-bones in the museum itself which are nearly of cised a paramount influence on these arts. It was these dimensions; such an error implies a want of at this period that a definite separation of pharmacy cooperation between the zoological and palæontofrom medicine first took place.

logical departments of the museum. Of less importEach of the succeeding chapters deals with the

ance, although far more embarrassing to the public,

is the discrepancy between the terminations of the progress made during a particular century, an out

orders ” of ostrich-like birds in the list on p. 8 and line of the additions to physical, chemical, and those in the synopsis on that and the following pages. botanical sciences being first given, with short bio- Again, we venture to think that the public will not graphies of the more famous exponents of these be likely to understand the semi-scientific jargon fresciences. The bearing of these discoveries on phar

quently employed in the text. The expression, for

instance, maceutical methods is then outlined, and finally the

on p. 106, “the remarkable Australian legislation of the periods, the social and commercial

forms constituting this order," would have been much

better had the word “birds " been used in place of conditions, and other matters in so far as they affected


Neither is the construction of the senthe practice of medicine and pharmacy are discussed. tences in all cases so good as it might be, as witness The book is evidently the outcome of much literary

the following (p. 64) :-“ The appendage opens under and antiquarian research on the part of its author,

the tongue and is largest in the male, giving the but it is unfortunate that more care was not exercised

bird a very peculiar appearance. Like its allies it is an expert diver . .

R. L. in selecting the material to be included. There is no reason why so much space should be taken up in re

A Laboratory Manual of Organic Chemistry for

Beginners. By Dr. A. F. Holleman. Translated counting the political and religious histories of the

by A. Jamieson Walker, Ph.D. Pp. xiv +78. (New various peoples. Similarly, the short and necessarily York: John Wiley and Sons; London : Chapman inadequate biographies of eminent men

en of science, and Hall, Ltd., 1904.) Price 4s, net. which are scattered broadcast through the second half The preparation and properties of a number of of the book, might well have been omitted, since they organic compounds are dealt with in short paragraphs are already better done elsewhere. By omitting these in a manner reminding one of the text-books of and other not strictly relevant matters, the size of the condemned. But little attempt is made to indicate

qualitative analysis, which are now so universally volume could have been much reduced, and at the same the quantities which should be used, and no emphasis time it would have been unnecessary for the author whatever is laid on the importance of making organic to write in the compressed, unreadable style which preparations in a quantitative manner. We even now characterises the book. As it is, the volume can

doubt whether the beginner would attain the required only be regarded as a useful work of reference on

result in performing many of the preparations de

scribed. the history of pharmacy and allied subjects, and to

It will be a sad day for the future of organic this purpose its index (26,000 entries) is well adapted. chemistry if text-books such as Dr. Holleman's come

T. A. H. into general use; it is indeed difficult to imagine

anything more calculated to encourage scamping of

laboratory work. OUR BOOK SHELF.

A growing complaint of the

chemical manufacturer abroad at the present time is Guide to the Gallery of Birds in the British Museum. that the university graduates from the large modern

Pp. iv +228; illustrated. (London: Printed by laboratories are ruined by the elaborate apparatus, Order of the Trustees, 1905.) Price 25. 6d.

ready-made reagents and other time-saving appli. This handsome volume is a new departure in the ances placed at their disposal, so that they are no matter of " guides," so far as the natural history longer themselves capable of facing practical problems branch of the museum is concerned, being larger in properly or of making the best use of the ordinary size, more fully illustrated, different in style, and technical appliances. The physical chemical epoch (perhaps most important of all) higher in price than I from which chemical science is now slowly recover

ing has caused it to be forgotten that for successful confusion; for instance, the two German words work in chemistry it is essential that the investigator | Eiweisskörper and Albumine are both translated as be a highly skilled manipulator. It is too often found albumins. that the best student in the examination room is all The work is primarily intended for students, and but worthless when set to periorm even the simplest therefore references to literature are omitted. A picce of experimental work ; good workers can only desire to keep the book within a moderate compass be trained by the most careful and thorough ground-, has no doubt induced the author to leave out a coning in making pure chemical preparations and by i sideration of many subjects which might well have being taught to appreciate the importance and been expected to find a place in it. Thus we find no ncossity of even the minutest details in the process. reference to the important subject of immunity and As a glance at the modern chemical literature shows, its side issues, like the precipitin test for blood. The it is precisely this attention to detail which is so numerous investigations now in progress on the conspicuous a feature in some of the best work.

velocity of ferment action are passed over in silence. We fear that the book under notice would not | Physical chemistry has during the last decade made lead the student to attach importance either to great progress, and many and important are its appliaccuracy of method or to thoroughness of detail; it cations to physiology. Such questions as absorption, seems a pity even that it should have been found secretion, osmosis, gaseous exchanges, and electrical worth while to translate it and so add another to the conductivity have all been made clearer by the work legion of text-books.

of the physical chemist; but there is no reference to

any of such investigations. Metaphysik in der Psychiatrie. By Dr. P. Kronthal. The strangest and most important omissions, how

Pp. 92. (Jena : Gustav Fischer, 1905.) Price 2.50 ever, are the absence of any account of general metaimarks.

bolism, animal heat, and respiration. THIS costly little work is written to ventilate a Turning to the title-page, one searches in vain for grievance. It would appear that certain authorities the words vol. i., for the omitted material would easily on mental diseases, including Kräpelin and Bins- / fill a second volume of the same size. One cannot wanger, employ in their works such terms as help thinking that, interesting and instructive as the association, apperception, power of imagination, book undoubtedly is, it cannot be expected to take its anger, and the like. These, according to our author, place as a favourite until the deficiencies alluded to are metaphysical terms, and must be carefully ex-l are rectified. cluded from Psychiatrie, which is a purely natural science. Sew sciences spring up like mushrooms Astronomy for Amateurs. By Camille Flammarion. nowadays, and it is a misfortune that those who Translated by Frances A. Welby. Pp. 340. (London : specialise in one, or seek to exploit it, so rarely know

T. Fisher Unwin, 1905.) Price 6s. with precision what is being done in others, even Much that is interesting to amateur astronomers, may when these are most closely akin to their own darling | be found in this volume. The descriptions are often pursuit. We fear that this writer hardly understands discursive, but the matter is there, and in a readable that the terms which he criticises are used every day form providing the reader's leisure is not too limited. in psychology with a minimum of metaphysical refer After a general exhortation to his readers to study ence, and that he is almost bound, before he proceeds and contemplate the marvels of the sky, the a step, to show due cause why the terminology of ceeds to a study of the constellations, the stars themPsychiatrie should differ seriously from that accepted selves, the sun, and then the planets. Next follows a by ordinary psychology. In spite of his parade of foot chapter on comets, containing some interesting facts notes and his references to such grand conceptions as concerning the ancient ideas of these “ glittering, that of Albeseeltheit, it may be doubted if this writer swift-footed heralds of Immensity," and a brief is competent to discuss so general a question. It any account of comets in general and of a few in particular. rate, his present work does not impress one as being Shooting stars are then dealt with, and in chapters well arranged, clear, or convincing.

viii., ix., and x. the earth, the moon, and eclipses are

severally discussed. In chapter xi. the more elementary 1 Text-book of Physiological Chemistry. By Charles methods of determining stellar distances and masses

E. Simon. Second edition. Pp. Xx + 300. (Lon are described, whilst the next, and last, chapter is dedon: J. and . Churchill, 1903.) Price 155. net. voted to a discussion of life universal and eternal. ALTHOUGH Dr. Simon's book has reached a second The book contains eighty-four illustrations--the reedition, it is one which has been hitherto un levance of some of which is open to question--and it known on this side of the Atlantic. Dr. Simon's will be read with both interest and profit by those name is not associated with any researches in physio whose previous acquaintance with astronomical truths logical chemistry, and there is nothing strikingly new has been slight. or original in his book, either as regards subjectinutter or arrangement. The work has, however, many excellent features. It is clearly written, and is

LETTERS TO THE EDITOR. free from inaccuracies; the sections dealing with the The Editor does not hold himself responsible for opinions proteids and their cleavage products are especially expressed by his correspondents. Neither can he undertake good, and fully abreast of the recent advances which to return, or to correspond with the writers of, rejected have been made in this important and interesting manuscripts intended for this or any other part of NATURE. branch of the subject.

No notice is taken of anonymous communications. ] The author is conversant with chemical technique,

Scientific Correspondence of the late Sir George Stokes, and his descriptions of analytical processes are specially lucid. It is evident that he is a careful

ARRANGEMENTS are in progress for the publication of a student of chemico-physiological literature, and more

selection from Sir George Stokes's scientific correspondence.

| The letters addressed to him, which are now in my custody, especially with that part of it which originates in

show that there must be many from him to others, of Germany. This is frequently seen in the nomen

permanent scientific value, to which I have not access. I clature he adopts. Thus he speaks of casein and shall therefore be glad if owners of letters of substantial para caspin instead of caseinogen and casein re scientific interest will entrust them to me, to be treated spectively as employed in most English books. | with care and ultimately returned.

J. LARMOR. Occasionally the adherence to German tcrms leads to St. John's College. Cambridge, May 8.



The Transposition of Zoological Names. the five compound singularities called the tacnode, the I to say how thoroughly I agree with Mr.

rhamphoid cusp, the oscnode, the tacnode cusp, and the Lydekker in his remarks on the unwisdom of transposing triple point. Also, since an evectant is the tangential zoological names, and on the confusion caused by this equation of a curve which is related in a special manner

to the original one, an examination of the evectants of objectionable practice. To the instances which he has mentioned I may add the following cases relating to two

the above quantic would lead to interesting results conwell known and familiar species of animals. Linnæus cerning conics and other curves connected with trinodal called the only European hare known to him Lepus

quartics. timidus, and for many years that name was applied to the

In this subject geometrical methods are a powerful common brown hare of Central Europe, while the northern

assistance to pure analysis. For example, let U be a hare, which changes to white in winter, was known by ternary cubic in (a, b, Y); eliminate y by means of the Pallas's appropriate name, Lepus variabilis. This

equation B=kay, and equate to zero the discriminant of the the nomenclature used by Blasius, by Bell in his

British resulting cubic equation in a B. This will give a sextic Quadrupeds,” and in all the ordinary text-books of zoology.

equation A (k)=0, which determines the six tangents drawn It was, however, pointed out some years ago, first, I

from A to the curve. The condition that the curve U=0 believe, by Lilljeborg, that the Lepus timidus of Linnæus

should have a node is that the equation A (k)=0 should had been based mainly upon the northern or variable hare,

have a double root; hence the discriminant of this binary or that at all events Linnæus had confounded the two

sextic is the discriminant of the original ternary cubic C. species together. In these circumstances obviously the

Many other examples of a similar kind could be menbest plan was to call the middle-European brown hare by

tioned, and we may observe that from the discriminant its next given name, Lepus europeus, and this course has

of a binary duodecimic, all the conditions that a quartic been adopted by most writers. But the advocates of un

curve should possess point singularities may be obtained. restricted priority are not content with this, and insist

April 28.

A. B. BASSET. upon calling the variable hare Lepus timidus, the consequence being that when that name is used it is impossible

Current Theories of the Consolidation of the Earth. to know which of two perfectly distinct animals is intended by it.

IN Lord Kelvin's philosophical and justly celebrated Another still more objectionable transposition of two

paper on the secular cooling of the earth (Thomson and well known names has been lately suggested. Linnæus, in Tait's “Nat. Phil.,” vol. i., part ii., Appendix D), the the twelfth edition of the “ Systema Naturæ,” gave the assumption is made that the earth was once a fiery molten naine Turdus musicus to the song-thrush and that of mass, liquid throughout, or melted to a great depth all Turdus iliacus to the redwing, and these familiar terms round. He cites Bischof's experiments showing that have been used by all writers for these well known birds “ melted granite, slate, and trachyte all contract by somerespectively ever since. But about a year ago it was dis- thing about 20 per cent. in freezing," and continues :covered by an ardent member of the new school of priority Hence, if, according to any relations whatever among that in his tenth edition of the “ Systema” Linnæus had the complicated physical circumstances concerned, freezing unfortunately (by some error in his MS. or of his printer) did really commence at the surface, either all round or attached the diagnosis of Turdus musicus to T. iliacus, and in any part, before the whole globe had become solid, the that of T. iliacus to T. musicus. It was admitted that

solidified superficial layer must have broken up and sunk Linnæus had corrected the mistake in his later edition of

to the bottom, or to the centre, before it could have 1760, but even Linnæus could not be allowed to correct attained a sufficient thickness to rest stably on the lighter his own errors in the face of the inviolable law of

liquid below. It is quite clear, indeed, that if at any time “ priority.” In future, therefore, it was maintained, the the earth were in the condition of a thin shell of, let us song-thrush must be called T. iliacus and the redwing

suppose, 50 feet or 100 feet thick of granite, enclosing a T. musicus ! This course has been actually adopted by a continuous melted mass of 20 per cent. less specific gravity subsequent writer, but we may trust that it will not meet

in its upper parts, where the pressure is small, this conwith general approval, and that the song-thrush and red

dition cannot have lasted many minutes. The rigidity of wing will remain under the old names given to them by a solid shell of superficial extent so vast in comparison the father of scientific nomenclature in 1760, and used by with its thickness, must be as nothing, and the slightest every subsequent writer until 1904. P. L. SCLATER.

disturbance would cause some part to bend down, crack,

and allow the liquid to run over the whole solid. The Modern Algebra.

crust itself would in consequence become shattered into

fragments, which would all sink to the bottom, or meet The publication of Messrs. Grace and Young's treatise in the centre and form a nucleus there if there is none to on algebra will direct attention to the importance and begin with.” difficulty of the theory of the concomitants of ternary and In adhering to these views, Lord Kelvin has been quaternary quantics in connection with plane and solid followed by Prof. G. H. Darwin (cf. “ Tides and Kindred geometry. There are one or two points on which I propose Phenomena of the Solar System, p. 257) and other to make some remarks.

eminent mathematicians; so that the theory that the earth In the first place, canonical forms are sometimes de- consolidated by the building up of a solid nucleus through ficient in generality, and this will be the case whenever

the sinking of portions of the crust of greater specific the form is the analytical expression for some special gravity is no doubt generally accepted by geologists and property of an anautotomic curve. Of this defect the canonical form of a

others interested in the physics of the earth. ternary cubic furnishes a striking Recent researches on the pressures within the planets example, for it is the analytical expression for the theorem (cf. Astronomische Nachrichten, No. 3992) have thrown that through each of the three real points of inflexion one great doubt on this mode of consolidation of the globe. real straight line can be drawn which passes through one The line of argument by which we reach this conclusion pair of conjugate imaginary points of inflexion on is a double one :anautotomic cubic curve ; and since autotomic cubics do (1) It is shown that the effect of pressure in the highly not possess this property such curves cannot be represented heated fluid assumed to have constituted the molten earth by the canonical form.

would have been to dissolve the portions of the sinking In the next place, anautotomic curves are not by any crust before they attained any considerable depth. means the most interesting species of curves, and to go (2) The increasing density of the fluid itself would have through the process of calculating their concomitants, and

prevented sinking of the crust below one-tenth of the then specialising them for some particular species of auto- radius, so that a solid central nucleus could not have been tomic curves, is often very laborious. In the case of

built up in this way. unicursal quartics, nany interesting results might be To see this clearly. let us suppose that the earth were obtained by calculating directly the concomitants of the

a molten mass, and that a crust of rock several kilometres quantic ( +0By, ya, ), and this would give results applic- in area, and a considerable fraction of a kilometre in thickable to all unicursal quartics, except those which possess ness, had formed, and begun to sink in the molten Auid


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