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In the condensed edition the cartouches of Egyptian amateurs about the principles that underlie the conkings which stood at the head of the chapters in the

struction of photographic lenses. second English edition have been placed at the beginning

The present work will serve as an excellent guide to

those who wish to gain this information, and should be of the book, and Miss Brodrick has added five pages of found to be of great practical use. The author has dealt matter on the Dêr el-Bahari mummies.

with the subject in a very popular manner, and although We have long hoped that Dr. Brugsch would issue a the mathematics is reduced to a minimum, he has made new edition of his “Aegypten unter den Pharaonen," his meaning very clear throughout.

In the first few chapters the nature and properties of revising his facts in some places, and correcting his

light are discussed, together with explanations of photostatements in others, and also adding the new facts

graphic definition, single and achromatic lenses, cause of relating to the periods between the VII.-XIth and XIII.- the inverted image, spherical aberration, nature and funcXVIIth Dynasties, which have recently come to light ; tion of diaphragms, nature and cure of distortion, optical failing this, which is much to be desired, we hoped that centres of single and combination lenses, &c. Chapone of his pupils would do the work under his guidance. ters, xi. to xv, treat solely of lenses, including accounts That, however, the English translation made by Seymour copying, and universal lenses. As there are thirty-nine and Smith, mutilated and robbed of its notes, and of the chapters in all, we may mention that of those remaining additions of the author, should be issued as a popular there are many on subjects which may be of special text-book of Egyptian history under Brugsch's name is interest to individual readers. Thus we have a chapter a fact which we deplore.

dealing with photo-telescopic lenses, a short one on the grinding of lenses, and another on enlarging and projecting in relation to lantern optics.

It will be seen that the author has dealt with a OUR BOOK SHELF.

wide range of subjects in which the lens makes its ap

pearance, and the reader will find that the explanations The Story of the Hills : a Popular Account of Mountains, are lucid, while the illustrations bring out the points and how they were made. By the Rev. H. N. Hutchin

which they are intended to show with equal clearness. son, B.A., F.G.S. (London: Seeley and Co., 1892.)

W. This is a pleasant, chatty book, all the more welcome because wholly unpretentious ; not too deep for “human The Evolution of Life ; or, Causes of Change in Animal nature's daily food " when roaming among the hills of Forms. A Study in Biology. By Hubbard Winslow which it treats. It will be read with pleasure and profit Mitchell, M.D. (New York and London: G. P. by the tourist, who likes to know just enough about the Putnam's Sons, 1891.) sundry points of interest connected with the scene of his wanderings to make the enjoyment of his outing intelli

DR. MITCHELL says in the preface to this book that he gent, but who is not haunted by a feverish anxiety to be has accomplished in it “all that can be reasonably exfor ever, in season and out of season, improving his pected from a medical man deeply immersed in the duties mind. Many who would shrink from a formal scientific

of his profession.” What most people expect from medical treatise with horror or disgust will find themselves able

men in this position is that they will not write books on to enjoy this book, and through its channel scraps of use.

vast and complicated subjects, for the proper treatment ful knowledge inay insinuate themselves into their minds

of which an author must have not only exceptional ability which would never have found their way there by any

but ample opportunities for philosophic study. So far as other road.

we have examined the work, it has neither freshness of Part I. is multifarious, and touches on a vast variety thought nor charm of style. Dr. Mitchell mentions that of matters more or less connected with mountains, and

he has travelled in many different parts of the world. If principally of human interest-mountain races, mountain

he was determined to write a book, he would have been legends, the uses of mountains to mankind, mountain better employed in recording his reminiscences as a storms, avalanches, and the plants and animals of moun

traveller than in tediously discussing questions which tains. Scientific explanations of facts and phenomena have occupied so many of the foremost intellects of the are interspersed : the severe critic may detect a little present age. vagueness and looseness here and there in these, but no very serious lapse. Well-chosen quotations from Ruskin and other authors give brilliancy to the narrative. There

LETTERS TO THE EDITOR. are landscape views reproduced from photographs, which have all the excellences and the artistic failings of this [The Editor does not hold himself responsible for opinions exclass of illustration.

pressed by his correspondents. Neither can he undertake Part II. is mainly taken up with a geological history of to return, or to correspond with the writers of, rejected mountains. Here all the main geological truths that

manuscripts intended for this or any other part of NATURE. bear on the subject are expounded clearly, and with

No notice is taken of anonymous communications. ] great fullness of detail. In fact, an epitome is given of a large number of the leading doctrines of geology, which

A Magnetic Disturbance. will suffice for the needs of many a general reader. A Our attention having been directed for some days past towards separate chapter is devoted to volcanic mountains and a spot of unusual size upon the sun's disk, we were not by any volcanic activity. We may note that the three stages in means surprised to observe, as doubtless many of your readers the life of a volcano mentioned on p. 266 are not such as

elsewhere also did, an aurora of great beauty on Saturday night are usually defined by geologists.

A. H. G. last ; nor was our anticipation of seeing a magnetic disturbance

portrayed upon the magnetograph records disappointed in the The Optics of Photography and Photographic Lenses developed, we saw that perturbations more violent than any

morning, for when the sheets were changed and the photographs By J. Traill Taylor. "(London : Whittaker ani Co., which had been recorded at Kew for the past ten years had 1892.)

been in progress since about 5.45 a.m. of February 13. ALTHOUGH photography is so widely practised at the The magnels were very quiet on Friday, but early on Satur. present day, fit is surprising how little is known by day morning they became disturbed. The easterly declination slightly increased until about 5.40 p.m., whilst both horizontal No doubt in many ways the treatment he suggests would have and vertical forces similarly increased in intensity, more espe.

been better. It would, for instance, have enabled me to prove cially between 4 and 6 p.m. They further diminished in force the case a fortiori. Perhaps, however, the reasons given in after 10 p.m., and their changes became very rapid from 12 the chapter explaining “why the book bas been written" may midnight to 2 a.m., whilst at the same time the declination show that for the object I had in view the method actually used proceeded to its extreme westerly position. Subsequently, the was appropriate. fluctuations in magnetism became much reduced in extent, and I am also much obliged to the same friend for pointing out the whole disturbance gradually diminished and died out about that the astronomical theorem proved in the appendix had 4 p.m. of Sunday.

been given by Wiener, Über die Stärke der Bestrahlung The Kew magnetometers were not able to record the complete (Zeitschrift der Oesterreichischen Gesellschaft für Meteorologie, extent of the vibrations to which free needles were subjected, vol. xiv., 1879, p. 129).

ROBERT S. BALL. nor could the entire change of force be secured in the field of Observatory, Co. Dublin, February 9. the instrument. The limits, however, clearly recorded were 2° of declination from '1760 to '1830 of horizontal force, and from 4350 to 4420 units of vertical force expressed in C.G.S. measure in absolute force. G, M, WHIPPLE,

Ice Crystals. Superintendent. WITH reference to the letter on the subject of ice crystals Kew Observatory, Richmond, Surrey, February 16.

which appeared in NATURE of the 4th inst. (p. 319), it is

perhaps worth mentioning that a paper on the subject, entitled The New Star in Auriga.

** Eine Eiskrystallgroite," by C. A. Hering, appeared in Groth's

Zeitschrift für Krystallographie und Mineralogie, Band xiv. PROF. COPELAND has suggested to me that as I am the (1888), pp. 250-253, and Plate vi. writer of the anonymous postcard mentioned by you a fortnight The crystals occurred in an old mine on the Waschgang near ago (p. 325), I should tell your readers what I know about the Döllach in Carinthia. Large fans, as much as 300 mm. long. Nova.

200 mm. broad, of ice-crystals grow out horizontally from the It was visible as a star of the fifth magnitude certainly for vertical walls. The stalk, consisting of a series of hexagonal two or three days, very probably even for a week, lefore Prof. prisms, hollow, like thermometer-tubes, was in the middle Copeland received my postcard. I am almost certain that at 25 mm. thick and thickened towards the point of attachment to {wo o'clock on the morning of Sunday, the 24th ult., I saw a the rock. The fan surface was a large hexagonal plate with bfth magnitude star making a very large obtuse angle with B strong prismatic ribs running from the centre to the angles. Tauri and x Aurigæ, and I am positive that I saw it at least The interspaces between the ribs were filled by prisms arranged twice subsequently during that week. Unfortunately, I mistook with the greatest regularity. Upon the ribs of the fan either it on each occasion for 26 Aurigæ, merely remarking to myself single crystals or funnel-shaped structures with step-like sides that 26 was a much brighter star than I used to think it. It consisting of prisms were borne. The individual crystals were was only on the morning of Sunday, the 31st ult., that I satisfied almost all thick tabular forms, with prism, basal pinacoid, and myself that it was a strange body. On each occasion of my rhombohedral faces.

BERNARD HOBOX. seeing it, it was slightly brighter than x. How long before the Owens College, Manchester, February 8. 24th ult. it was visible to the naked eye I cannot tell, as it was many months since I had looked minutely at that region of the heavens.

A Rare British Earthworm. You might also allow me to state for the benefit of your readers that my case is one that can afford encouragement to In the summer of 1890, during my researches into the even the humblest of amateurs. My knowledge of the techni- | Vermes of Cumberland, I discovered a species of earthworm calities of astronomy is, unfortunately, of the meagrest descrip which proved to be new to Britain (Lumbricus Eiseni, Levinsen). tion; and all the means at my disposal on the morning of the As I have recently had the good fortune to receive specimens 31st alt., when I made sure that a strange body was present in of the same worm from another part of the country, it seems ibe sky, were Klein's “Star Atlas," and a small pocket tele- desirable to place the same on record. A correspondent writes scope which magnifies ten times.

from Gloucestershire as follows :

THOMAS D. ANDERSON. “ Last Saturday (January 30, 1892), I walked up to one of 21 East Claremont Street, Edinburgh, February 13.

my favourite woods here on the Cotswolds, about 700 feet above the sea-a damp old beech wood, the Frith Wood of Withering's

Arrangement," seventh edition, 1830—and seeing a stump of Nacreous Clouds.

some 10 inches diameter with a growth of the black · Candle In the morning of the 30th ult. there was a magnificent display Snuff Fungus'on it, I examined the rotten wood, which gave way of the nacreous (or iridescent, as they were first called) clouds, to the pokes of my stick. Among this rotten wood I saw some which formed such a striking feature of the sunset and sunrise earthworms, two or three of which I inclose, hoping they may sky for some days in succession in December 1884 and 1885 prove an addition to our worm fauna." rol. xxxi. pp. 148, 192, 316, 360, &c.). They were not exactly I have placed on record all the known earthworms of the same in appearance, but I should say they were of the same Gloucestershire in The Field Club for 1891, to which this may nature. I had not seen them in the interval of six years, and now be added. The worms were small, but in good form for have only noticed them lately on the one day mentioned. They identification, and prove to be specimens of Eisen's worm. I were confined to the southern part of the sky. As the sun rose have, unfortunately, been unable hitherto to consult Levinsen's higher their colours were less visible, and the clouds disappeared original description; nor have I been able to obtain Rosa's about noon; though in the afternoon some reappeared, but memoir published in the Boll. Mus. 2001., Torino, 1889 (vol. iv., dever became very striking. At 5h. 44m. G.M.T. there was No. 71). I am therefore obliged to content myself with a only one group, which was too far from the sun to show any description of the specimens in my possession. nacreous colours ; its centre was about at bour-angle ih. 2m. Lumbricus Eiseni, Lev., as found in Britain, is a small species west, and declination 23° south. Although conspicuous they of earthworm, measuring about 15 inches in length when adult. were no longer very bright, and I should say the sun It has the usual colour of the allied species-the purple and evidently not shining on them, for they were the same bluishred worms-being of a ruddy hue, with iridescence. The green colour as the western sky, and I apprehend were clitellum or girdle, which occupies segments (24) 25 to 31, is a illuminated by the sky.

T. W. BACKHOUSE. reddish brown, being lighter in colour than the anterior portion Sunderland, February 9.

of the worm's body on the dorsal surface. Ventrally the worm

is, as usual, of a lighter shade. No tubercula pubertatis have The Cause of an Ice Age.

been seen under the girdle, but the first dorsal pore in every

specimen examined is clearly detected behind the 5th segment. In his very kindly review Prof. Darwin thinks I might have This may be indicated by the fractional sign; and as the most stated my argument with more completeness if I had preserved recent researches tend to demonstrate the constancy of this its generality by the use of a symbol instead of taking a special character for each species of earthworm, it is important to note case.

the same. The lip or prostomium has tha complete mortise and

was

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28 to 33.

tenon arrangement which distinguishes Lumbricus from Allolo- view, mathematical formulæ could ever have been construcied. bophora.

"A mathematical formula," Mr. Dixon remarks,“ does not

imply the existence of any instance whatever of its application, -pros.

any more than a definition implies the reality of the thing pros.

defined." But if a definition is always of a thing, what more is -peris.

wanted ? The definition is admitted to be of something ; and -peris.

what is something must, I suppose, exist somehow.

(4) I still think that in the passage in Mr. Dixon's letter which I referred to under (4) he is not consistent. For if, as he asserts, the definition of four as = 1+1+1, makes it false to say

that Twice two are four, this is surely because the facts referred The male pore is situated normally

on segment 15, but as the to by jour are no longer what they were when the statement in papillæ which carry the pores are large, they extend over the question was true. If definitions were purely arbitrary, as Mr. adjoining segments on either side. Earthworms vary greatly in Dixon holds, what would prevent my saying that Four (1+1+1) this respect. Rosa says that spermai hecæ are absent in this species, a peculiarity which has been noted in worms belonging

means twice two (1 + 1) + (1 + I)? It is surely only the reserto several other genera. I have not sufficient material to enable

ence to things which makes it absurd—and, however four (4) me to confirm or dispute this statement at present

. I have may be defined, how is one (1) to be understood, except by refercounted the segments of three specimens, and found them to be

ence to things ?). in each instance 106.

That words and symbols used intelligibly do, and must, refer As the year advances I hope to be able to obtain mature adults for dissection, when it will be possible they did not, no assertion of the form S is could ever be

to something beyond themselves, seems to me indisputable. If to give a detailed account of the internal anatomy. Meanwhile made, for the symbol S is certainly not the symbol P. And for the external characters are amply sufficient for distinguishing the worm if the girdle is properly developed, as its nearest British any statement, of the form Sis P, to be possible and significant, ally (Lumbricus purpureus, Eisen) has the clitellum on segments cation, but diverse signification.

it is further necessary that S and P should have identical appli

If application and significaHILDERIC FRIEND.

tion were the same, we should get S is S and Pis P; if applicaIdle, Bradford.

tion were not the same, we must say, S is not P. Hence, no term

can ever be taken in mere denotation (or application), nor in The Implications of Science.

mere connotation (signification); but both momenta of each Will you allow me to say something in answer to Mr.

term have to be taken into account in every assertion. If (to Dixon's letter on this subject in NATURE of January 21 (p. 272) ?

take a case given by Mr. Dixon in his “ Essay on Reasoning,"
p. 8) we
" define

metal as (1) I admit that there is a verbal or symbolic 'convention"

"the list of denotation, iron, if two (or more) persons agree to understand any given words or

copper, tin, zinc, lead, gold, and silver," then iron, &c., can symbols in a way arbitrarily chosen by themselves. But the only be pointed out by taking some specimen of iron, and scope of such convention is exceedingly limited : if people wish saying, This and all other things which are like it in certain to be understood, or even to understand themselves, they must

respects. An absolutely arbitrary denotation can be given only use the same words as others use, and use them in the same if the whole of the objects denoted are severally pointed out; sense (except in an infinitesimal proportion of case-). If it is

and even then, unless they are labelled, they can only be resaid that the common application and use of current words is a

membered and identified by means of their characteristics ; if mere convention, the word convention is taken in an extremely

labelled, by that characteristic. strained and metaphorical sense, since nothing like an explicit

Mr. Dixon objects to my attributing to him the view that agreement has ever been made. The “convention" as to the

“mathematical truths in as far as 'real' are obtained by inducuse of language is as fictitious as the social contract of Locke

tion, and are therefore not necessary.” But in his letter of and Rousseau. But in the one case, as in the other, there is a

December so he says :-"For example, the assertion. Two solid basis of facts, to suit which the hypothesis has been pro- straight lines cannot inclose a space' is certainly not a 'necesduced. Language bas been moulded by thought and feeling, its truth depends solely on those definitions, or else its terms

sary truth. Either its terms are defined hy connotation, so that which, in their turn, have been impressed by facts ; and it is facts and relations of facts that language seeks to express.

are defined by denotation, as representing real things in space ;

As Mill says (in the first chapter of his “ Logic ") names are a clue to

and the truth of the assertion can only be proved by induction things, and bring before us “all the distinctions which have from actual experience with those things. In the first case, the been recognized, not by a single inquirer but by all inquirers might conceivably be false, as was shown by Helmholtz.” It

truth is arbitrary, not necessary; and in the second case it taken together." No one, I imagine, would say that a particular case of the impossibility of affirming and denying a

was this passage which led me to the opinion which I expressed. given statement, depends “solely on the law of contradiction”;

Cambridge, January 31.

E. E. C. JONES. but in the case of any particular assertion, the impossibility, in that case, is seen, and to a mind that has reached the generalizing

Vacuum Tubes and Electric Oscillations. stage, the universal is discernible in the particular. As regards the question of “real propositions,” I will not occupy space

I HAVE not had the advantage of hearing the lecture of M. with quotations, but will only refer to Mr. Dixon's letter of Dec.

Nikola Tesla nor of seeing his experiments, but it does not seem ember 10, in which the passages occur which led me to think

out of place to recall the attention of your readers to an article that he regarded assertions (or denials) of the existence of by Dr. Dragoumis in your issue for April 4, 1889, in vol. xxxix. particular objects as the only "real" propositions.

OLIVER J. LODGE. (2) As regards induction, I agree with Mr. Dixon that the starting point in induction is hypothesis or discovery. But with reference to the rest of the procedure, and its relation to

THE NEW STAR IN AURIGA. so-called “formal ” logic, I differ from him. For I think that an inductive generalization may be set out syllogistically ; e.g.,

SINO

INCE our last article was written the weather has What has once produced X will always produce X ;

continued very bad for astronomical observations. A has once produced X;

The only new results obtained which have reached us .. A will always produce X (= all A is X).

consist of a paper read by Mr. Norman Lockyer at the

Royal Society on Thursday last, and an important If space allowed, I should like to consider the justification for the telegram from Prof. Pickering, which appeared in major premiss, and also to say something about the grounds on Wednesday's Standard. which the minor (which indicates the hypothesis or discovery)

We will take these in order. Mr. Lockyer's comasserts causation (or concomitance) in a given instance.

(3) Mr. Dixon says : “We do not, in mathematics, conclude munication to the Royal Society was dated February 8 ; a universal proposition from a single concrete instance."

But it stated that two more photographs, containing many it appears to me that, as far as my own experience goes, in every

more lines than the former ones, were taken on Sunday concrete mathematical proposition which I understand this is night, February 7, and it went on to make the important exactly what bappens ; and I do not see how, on Mr. Dixon's announcement that “ The bright lines K, H, h, and Gare

p. 548.

sides."

Int Photo. Date, February

A obtained by means

of a curve.

and Photo. Date, February 3. A obtained by means

of a curve.

3rd Photo. Date, February 7. A obtained by means

of a curve.

3rd Photo.
À obtained by means
of a direct compari-
son with lines in

a Cygni.

Bright-line stars.

Orion stars
(dark lines).

Nebula in Orion

(bright lines).

accompanied by dark lines on their more refrangible falling off of intensity at the edges, as in the case of the

hydrogen lines in such a star as Sirius. With the method This was substantially the substance of the telegram employed in taking the photographs, long exposures are which appeared in the Standard on the following Wed- liable to result in a tbickening of all the lines on account nesday (February 10), with the additional remark that

atmospheric tremors. The lines would also be thick if the Harvard astronomers thought it possible that the the Nova be hazy. In the photograph, however, all the phenomena presented by the new star had been caused lines are not equally thick. by the collision of two celestial bodies.

If the lines are similarly broadened when a slit spectroOn the next day the detailed observations made on scope is employed, the effect must be due to internal Sunday night at Kensington, together with the approxi- agitations, for if different regions of the Nova are moving mate wave-lengths of the lines measured on the photo- with varying velocity, or with the same velocity in different graphs, were sent on by Mr. Lockyer to the Royal Society. directions, a normally fine line might be widened in the From these we learn that the Nova on Sunday appeared manner observed in the photographs. to be slightly brighter than on February 3.

With regard to the bright and dark lines the paper With the 10-inch refractor and Maclean spectroscope, states as follows:C was seen to be very brilliant, and there were four very “A somewhat similar phenomenon has already been conspicuous lines in the green. Several fainter lines were recorded by Prof. Pickering in the case of B Lyra, and also seen, and a dark line was suspected in the orange. this has been confirmed by a series of photographs taken Mr. Lockyer noticed that some of the lines, especially the at Kensington. In this case the bright lines are alterbright one near F, on the less refrangible side, appeared nately more or less refrangible than the dark ones, with to change rapidly in relative brightness, and this was a period probably corresponding to the known period of confirmed by Mr. Fowier.

variation in the light of the star. The maximum relaObservations of the spectrum were made by Mr. Fowler tive velocity indicated is stated by Prof. Pickering as with the 3-foot reflector and the Hilger 3.prism spectro- approximately 300 English miles per second. scope. These call for no special remark.

“In the case of Nova Aurigæ, the dark lines in all four Twenty bright lines have been measured on the photographs taken at Kensington are more refrangible photographs, and their wave-lengths are given in the ihan the bright ones, so that as yet there is no evidence accompanying table :

of revolution.

“ The relative velocity indicated by the displacement of Lines in the spectrum of Nova Auriga.

the dark lines with respect to the bright ones appears to be over rather than under 500 miles per second. The reduction is not yet complete.

“Should the photographs which may be obtained in the future continue to show the dark lines displaced to the more refrangible side of the bright ones, it will be a valuable confirmation of my hypothesis as to the causes which produce a new star-namely, the collision of two meteor-swarms On this supposition the spectrum of

Nova Aurigæ would suggest that a moderately dense 3933 3933 3933

3933 3933

swarm is now moving towards the earth with a great 3968 3968 3968

3970 3968 3968 velocity and is disturbed by a sparser one which is 4101

4101 4101 4101 4101 4101 4101 receding. The great agitations set up in the dense swarm 4128 4130 4127 4128

4130

4130 would produce the dark-line spectrum, while the sparser 4172 4172 4172! 4172

4172

swarm would give the bright lines." 4202 4200

4200 4226 4227 4226

4226 4268 4268 4264

4268 / 4268

ELECTRODYNAMIC THEORIES AND THE 4294 4291

ELECTROMAGNETIC THEORY OF LIGHT.1 4312

4310 4310 4310
4340 4340

4340 4340 4340 N a former article we endeavoured to give an account 4383

4383 of the first part of M. Poincaré's“ Électricité et 4412

4415 | 4410 Optique,” in which he dealt with the electric and 4434

magnetic theories expounded in Maxwell's treatise. In 4472, 4472 4472 Part II. he now compares the theory of electromagnetic 4522

4510 4554

action given by Maxwell with the somewhat more general 4555

4550 4584 4587

theory put forward by Helmholtz in his celebrated paper 4625 4625 4620

on the equations of motion of electricity (Pogg. Ann., 4860 4860 F 4860 4860 4860

cii. p. 529, or Wissensch. Abhand., vol. i.) ; discusses the condition which must hold in order that the two theories

may coincide ; and, after a masterly exposition of the The table also shows probable coincidences with the lines in the spectra of the Wolf-Rayet stars as photo-finishes with a very valuable analysis of the theoretical

various consequences which flow from Maxwell's theory, graphed by Prof. Pickering, dark lines in Orion stars and experimental work of Hertz. photographed at Kensington, and bright lines in the Orion nebula photographed at Mr. Lockyer's observa- of Ampère for the mutual action of two current elements.

In the first chapter M. Poincaré deals with the formula tory at Westgate. In addition to the lines recorded in the table, the principles assumed from Ampère's experiments :

The method adopted is founded on the following three photographs in the spectrum of the Nova show several lines more refrangible than K. They probably include

(1) That a current in a conductor may be replaced by

an equal current in a sinuous conductor nowhere some of the ultra-violet hydrogen lines. All the lines in the spectrum of the Nova are broad,

deviating from the first by a distance comparable with

the distance of the latter from any elemeat of the other although in a photograph of the spectrum of Arcturus, conductor acted upon. taken with the same instrumental conditions, the lines were perfectly sharp. It is also important to note that

Électricité et Optique." II. Les 'Théories de Helmholtz et les Ex

fériences de Hertz. Par H. Poincaré, Membre de l'Institut. (Paris : Georges the broadening of the lines is not accompanied by any Carré, 1891.)

K 3933

H 3968

4228

G 4340

4340

4469 4518

4516 4552 4587 4618

4516 4552 4587 4618

COS E

= SCos

+ B

ds

+ c da

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COS E 7

ተ 1

(2) The action of a closed circuit carrying a current The above expressions for F, G, H reduce easily to upon any current element is normal to the element. (3) The action of a closed (non-varying) solenoid

fr)dx'rdy'

)dz' so that, putting in the

upon a current element is zero.

value of f (r), we get the well-known value of the mutual It is besides assumed that the action of a circuit upon energy of the two circuitsa current element is the sum, in the dynamical sense, of the individual actions of the elements of the circuit ; and

T

dsds'. that the action between two elements is a force in the straight line joining their centres.

The theory of induction is next taken up. After a The process used for the deduction of Ampère's formula short discussion of some objections made by M. Bertrand from these premisses is very elegant. If'ds, ds' be the to the received method of deducing the laws of induction lengths of the two elements, y, y the currents in them, from the observed facts of electromagnetism, M. Poine the angle between the elements, 0, o' the angles they care proceeds to show that the electrokinetic energy of make with the line joining their centres, the action of ds two currents is equal to the electrodynamic potential, on ds' may be represented by f (r, 0, 0, e)yy'dsds'. But and recalls Maxwell's application of Lagrange's dynathe action of ds may, by the first principle stated above, mical equations to the theory of inductive action. He be replaced by the actions, of its components dx, dy, dz; then deals at some length with the celebrated theory put so that

forward by Weber for the action between two quantities, dx f dy

e, é', of electricity, as depending on their distance apart = A d's d's'

and their motion. where A, B, C are coefficients. Now, f depends upon discusses certain difficulties to which the theory leads in

This we pass over, with the remark that Poincaré here r, 0, 0', e; r and e' do not depend on the direction cosines of ds ; cos A and cos € are linear and homogeneous two current elements, and concludes with a short analysis

connection with the value it gives for the action between with respect to these direction-cosines. Hence f must be linear and homogeneous with respect to cos 6 and deduced from Weber's law.

of Maxwell's examination of the theory of induction as cos e, that is with respect to drds, and dor/dsds'. Simi

According to Maxwell larly, f is linear and homogeneous with respect to drļas', Weber's theory gives, for the inductive electromotive

(“El. and Mag.," vol. ii. p. 445, second edition), d’r dsds'. Hence we have

force exerted by the circuit in which the current y flows f = y(r)

ilr dr
+ 20(r)

on the other, the equation-
ds ds
dsds''

I dr dr where y(r) and o(r) are functions of r.

E

dsd's',

rds ds' These functions are determined by the second and third fundamental principles. The second gives ylor) which, for a closed circuit, agrees with the well-known

0(r), so that the problem is reduced to the determina- equationtion of p(r). This value of y(r), however, permits f to

d

E = be written in the form 2dU/dr.d’U/dsats',

M. Poincaré points out that this apparent agreement where U is a function of r only, and

of the two theories is due to the fact that Maxwell has

overlooked certain terms which contribute to the value dU/dr = Voir).

of E, and which do not give a zero result when integrated From this it is then shown that, if T be the so-called round a closed circuit. electrodynamic potential (electrokinetic energy) of the The expressions given by Weber and Neumann for the circuits-that is, the function the space variation of mutual potential of two current elements are next conwhich, for any direction, is the force in that direction sidered, and shown to be included in the general expresbetween the circuits

sion given for the same potential by Helmholtz. By dU dU T = dsds',

means of this expression Helmholtz's general electroas as

dynamic theory is introduced, and then follows an the currents being each unity, and the integrals being Maxwell. It is shown that Helmholtz's theory leads to

elaborate comparison of the theories of Helmholtz and taken round the circuits.

The determination of U is then effected by means of the value of T for conduction in three dimensions given the third principle. It is first shown that I may be by the equationwritten as the integral of Fdx + Gdy + Hdz round the

T = } /(Fu + Gv + Hw)da, circuit to which d's belongs, F denoting the integral round the other circuit of Uds'drlas'. (x − x)r, and where dw is an element of volume, u, v, w the components G, H similar expressions. F, G, H'are, in this theory, of currents, and the integral is extended throughout all what Maxwell has called the components of vector space. F, G, H, are, of course, the components of vector potential. These values of F, G, H, it is to be remarked, potential, and in this theory are given by, equationsfulfil the relation d F/dx + dGdy + dH/dz (= J) = 0.

fydx'lr + $(1 – k}/dx, &c., By applying the third principle it is proved that, if where p2 have its ordinary signification, and f'(r) U”?/r, (U' = dU/dr), of(r) must be a constant, in order that the action of a closed non-varying solenoid on a complete circuit may be zero.

If p be the density of free electricity at any point, Since f(r) must be zero at infinity, this gives f(x) = kr; and if the ordinary electro

dudx + dr'ídy + dw/dz - dpdt, magnetic definition of unit current be taken, k must and this, when instead of y'drfus' is substituted its value be unity, so that U' + 1/ Jr. Hence the attraction in terms of u', v, w, gives, by an application of Green's between the elements is

theorem, the result (cose - cos 0 cos e'),

rdp'ldt.cã, Ampère's well-known expression.

where dâ' is an element of the space in which the current

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F= 11

= jy'ds". dir/ds'.

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2yy'dsds'

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