In the first case, in substituting av for 1, and R(at) for Emu2, the well-known formula of Prof. Van der Waals is arrived at. In the second case, the same substitution leads to a quite worthless formula, unfit to explain even qualitatively the conduct of gases under compression.

The first form is the one which presents itself most naturally when, as was done by Van der Waals, the extension of the molecules is considered as a diminution of the volume in which they are moving; the second is obtained as a first approximation, when the virial equation is extended to the repulsive forces which come into play at the collisions. Of course, both methods, if they could be worked out with absolute rigour, would give the same result; but, this being impossible for both of them, the question as to which gives the better approximation is not at all an unreal one.

Now, it is extremely improbable that this question should have to be answered in a different way for linear and for three-dimensional space; yet for linear space the first method leads to a quite easy and absolutely rigorous solution, and the equation thus obtained is analogous to the first form.

In order to prove this, let A B (Fig. 1) represent a linear space of length 7, bounded by two rigid walls, A and B, and let there be moving in this space some perfectly elastic particles, all of the same mass, m, and length, X, but having different velocities, V1, V2, · Vu. At every encounter of these particles there will be simply an exchange between their velocities; therefore at every moment one of the particles will have the velocity v. On this particle we fix our attention, following it on its way till the next collision. After this collision we leave it to its fate, directing our attention to the other particle, which has now acquired the velocity v. Proceeding in this manner, it is obvious that at every collision a distance A is economized, which has not to be travelled over by the centres of the molecules. Starting, then, from the wall A and passing over to the wall B and back again, the number of collisions is 2(n − 1), and the distance economized 2nλ, (adding 2λ for the collisions against the walls). The distance travelled over by the centres consequently being 27 – 2nd, it is clear that the number of collisions with velocity v, against one of the walls in one unit of time, and the corre

amounts to

V1 2(L-nλ) sponding change of momentum to pressure on the wall is measured by

His paper, published in Wiedemann's Annalen, Bd. xii. p. 127, was inserted in the German and English versions of Van der Waals's pamphlet " On the Continuity of the Liquid and Gaseous States of Matter," at the end of the sixth chapter.

Considered as a determination of the factor, with which the total volume b1 of the particles, when introduced in vb, is to be multiplied, our results were identical, and confirmed the opinion expressed by Van der Waals about the value of this coefficient. Mr. Lorentz viewed his results in no other light, and had no intention at all to substitute his formula (5) for that given by Van der Waals. Indeed, in the passage of his paper which I quote here, he clearly indicates the weak point of his calculation :-"Strictly speaking, a correction ought to be made here, indicated by Mr. Van der Waals; in calculating the number of encounters, the extension1 of the molecules should have been taken into account. The matter is simplified, however, if the influence of the virial arising from the repulsive forces, or the size of the molecules, is small; and if a correction to the first order is sufficient, then the uncorrected value of the number of encounters may be used in calculating the small repulsive virial.”

Now it is not impossible to apply to Lorentz's formula the correction alluded to in this passage. In 1875 I calculated for the first time, by a more rigorous method, the shortening of the mean free path of spherical particles, in consequence of their extension in the direction of motion. Some months later, Mr. Van der Vaals succeeded in the same calculations by a somewhat different method, extending it to the case of two sets of particles of different diameters. Both calculations lead to the same result, viz. that the mean free path is shortened in the proportion v: v4b; therefore the number of the collisions, and the term in the virial equation dependent upon these collisions, must be augmented in the reciprocal proportion; but then this equation takes the form4b1 ข

mv12 L ηλ

so that the

pv = }±mu2 ( 1 +

=

= }Σmu2.

(6)

(3)

Of course, for space of two and three dimensions, the problem is much more complicated. Yet in 1877 I gave a solution of it for spherical particles which, according to my opinion, is rigorous so far as the several encounters between the molecules may be looked at as independent of one another. For a short time after each collision the possibilities of fresh collisions are considerably influenced by the proximity of the departing molecule. This influence, certainly of very difficult mathematical treatment, is disregarded in my calculations.

The outcome of these calculations is that of every

Verslagen en Mededeelingen der Kon. Ak. u. Amsterdam, 2o Reeks, Deel x.; Archives Neerlandaises, t. xii.

2 I am bound to acknowledge that the same correction, which is indicated further on for Lorentz's calculations, has to be applied to the number of collisions given in my paper.

and becomes identical with the equation (3) of Van der Waals.3

In this manner the true formula is obtained by means of the virial equation, as it has been by the method of economized distances, and these verifications of the equation derived by Van der Waals are not without importance. Indeed, I always held the opinion that it is not quite allowable to conclude directly from the diminution of the free path of the molecule to a proportional augmentation of the pressure on the bounding walls. The number of the mutual encounters of the molecules, and the number of their collisions with the walls (or, rather, their passages through an ideal plane), are not

The extension in the direction of the motion is meant here. I have translated the first phrase from the original paper in Wied. Ann., where it runs: "Streng genommen müsste man also hier eine Correction anbringen, wie sie von Herrn van der Waals angegeben würde; man hätte nämlich bei der Stosszahl die Grösse der Molecule zu berücksichtigen."

2 Verslagen en Mededeelingen, 2 Reeks, Deel x.; Archives Neerlandaises,

t. xii.

3 I owe this remark to a verbal communication by Van der Waals.

Lake Albert Edward.

THE death of Dr. F. C. Dietrich, Keeper of the Botanical Museum at Berlin, is announced. He was eighty-six years of age.

AT the annual general meeting of the Institution of Electrical Engineers on Thursday, December 10, Prof. Ayrton was elected President for the coming year. The following are the VicePresidents: Alexander Siemens, R. E. Crompton, Sir David Salomons, and Sir Henry Mance. In moving the adoption of the annual report, Prof. Crookes said that the number of members elected during the past year was greater than in almost any previous year. He announced that Prof. Nikola Tesla is on his way to England, and had promised to lecture before the Institution in January next. Prof. Crookes added that the Council would spare no pains to insure that the lecture should be thoroughly well experimentally illustrated. Mr. W. H. Preece, F.R.S. (Past-President), read a paper on "The Specification of Insulated Conductors for Electric Lighting and other Purposes." In this paper the fallacy of the present mode of specifying electric light conductors was exposed, and a new standard of insulation, based on the well-known qualities of gutta-percha, was proposed. The qualities of the numerous insulating materials now in the market were measured and determined in this new standard, and it was shown that any classification of cables should be based on the pressures to be resisted, and should depend on the thickness of the insulating wall. The introduc. tion of cheap and nasty cables, owing to competition and the absence of specification and inspection, was strongly commented on. It was shown that all danger was eliminated by the use of proper material and proper design. The paper concluded with the recital of Mr. Preece's latest specification.

PROF. A. HANSEN, of Darmstadt, has been appointed to the Professorship of Botany and Directorship of the Botanic Garden at Giessen.

Prof. E. WarninG, of Copenhagen, is at present engaged on a botanical expedition to the West Indies and Venezuela. Herr G. Schweinfurth and Prof. O. Penzig have returned from their journey in Abyssinia; and Herren J. Bornmüller and Sintenis from their botanical expedition, in the course of which they have visited the island of Thasos, Mount Athos, and the Thessalian Olympus.

THE following are the lecture arrangements of the Royal Institution before Easter, so far as they relate to science:-Prof. John G. McKendrick, six Christmas lectures to juveniles, on life in motion, or the animal machine; Prof. Victor Horsley, twelve lectures on the structure and functions of the nervous system (the brain); Prof. E. Ray Lankester, three lectures on some recent biological discoveries; Dr. B. Arthur Whitelegge, three lectures on epidemic waves; Prof. J. A. Fleming, three lectures on the induction coil and transformer; the Right Hon. Lord Rayleigh, six lectures on matter, at rest and in motion. The Friday evening meetings will begin on January 22, when a discourse will be given by the Right Hon. Lord Rayleigh, on the composition of water; succeeding discourses will probably be given, among others, by Sir George Douglas, Prof. RobertsAusten, Mr. G. J. Symons, Prof. Percy F. Frankland, Sir David Salomons, Prof. L. C. Miall, Prof. Oliver Lodge, Mr. John Evans, and Prof. W. E. Ayrton.

LAST week a deputation of gentlemen interested in the University Extension movement had an interview with Lord Cranbrook, President of the Privy Council, to ask for a Government grant in aid of the local lectures delivered under the auspices of

Stokes, Prof Bryce, Prof. Jebb, Mr. James Stuart, the President of Magdalen College, Oxford, the Master of University College, Oxford, and the Master of Selwyn College, Cambridge. Lord Cranbrook reminded the deputation that his official duties related only to public elementary schools, and that a Government grant could be obtained for the University Extension movement only from the Treasury. He expressed sympathy, however, with the objects of the movement, and promised to consider carefully and to bring before his colleagues the arguments advanced by the deputation. Referring to the general question of secondary education, Lord Cranbrook said it was most desirable that clever boys and girls who have passed through the elementary course should be enabled, by bursaries or in some other way, to go to intermediate schools, and thus be prepared for such instruction as is offered by University Extension lecturers. He feared, however, that those who expected this object to be attained by means of a Government grant might have to wait for some time."

ON the invitation of the Council of the Photographic Society of Great Britain, Mr. Leon Warnerke lately undertook to submit to the Society a description of the photographic technical schools on the Continent. With that object in view he visited, during last summer, Belgium, Germany, Austria, and Russia, taking notes with pencil and camera.

The results are embodied

in an interesting paper which was read at a recent meeting of the Society, and is now printed in the Photographic Journal.

THE organizing joint committee of the Essex County Council and the Essex Field Club on technical instruction have issued a circular announcing that they have resolved to appoint a certain number of lecturers on science subjects. The services of these teachers are offered free (with the exception of travelling and hotel expenses of the lecturers, where necessary) to local technical instruction committees, under certain conditions to be settled hereafter; the local committees guaranteeing audiences or classes of students (not less than twenty in number), providing rooms, gas, &c., and defraying all necessary local expenses. Syllabuses of short courses of lectures already approved are sent with the circular. They relate to elementary vegetable physiology, economic entomology, and elementary practical mechanics.

THE Royal Commission for the Chicago Exhibition are anxious to comply with a request made to them by the execu tive authorities of the Exhibition, that a typical collection of economic British minerals may be included in the British Section, and they are now applying to owners and managers of mines, asking for specimens of the principal British minerals. Mr. B. H. Brough, the Instructor in Mine-surveying at the Royal College of Science, South Kensington, has kindly undertaken to classify and arrange the collection, and any suitable specimens may be addressed to him. What is required is not specimens of special value or rarity, but samples of ordinary ores, &c., so that the collection when complete may be fully illustrative of the mineral resources of the kingdom. At the close of the Exhibition the collection will be presented to an American Museum

PROF. H. A. HAZEN, acting under instructions from the U.S. Weather Bureau, is in Chicago preparing a report on its weather-the mean temperature, the winds, snows, showers, humidity, early frosts and late nows. The report will be based on all the observations and records made for the last fifty years the object being to convince everyone interested in the approaching Exposition that Chicago is exceptionally favoured in point of fine weather.

MR. ROSEWATER, who was a distinguished member of the U.S. Military Telegraph Corp; during the American civil war, and is now President of the Old Timers Telegraphic Association, has lately been studying the various Government telegraph systems in use in Europe. The results of his investigations will shortly be submitted to what is expected to be an unusually interesting and important meeting of the New York Electric Club.

A COPY of "Whitaker's Almanack for 1892" has been sent to us a few days in advance of publication. Great care has been taken, as usual, to keep the Almanack up to date. Additional space is devoted to educational matters, and for the first time educational progress and occurrences are dealt with in a separate article. There is also a separate article on agricultural education. Other subjects separately treated are the rise, progress, and achievements of the great lines of ocean steamers, naval gunnery, and the results of the census. Of course, various sections resemble in subject those of former years, but even these are for the most part entirely fresh in substance. In many instances the changes wrought during the interval of a single year are so numerous that scarcely a line of the section in which the subject is treated remains unaltered.

OWING to his declining to take up his residence in Rio de Janeiro, Dr. Fritz Müller, of Blumenau, Sta. Catharina, has been summarily dismissed by the new Government of Brazil from his post of "Naturalista viajante" to the Museum at Rio de Janeiro. The great services which Dr. Müller has rendered both to zoological and botanical science during his forty years' residence in Brazil are too well known and too widely acknowledged to need dilating on. Dr. F. Müller is now close on completing his seventieth year; and Dr. Karl Müller, of Halle, the editor of Natur, proposes to seize the opportunity of collecting from his fellow-naturalists some substantial recognition of the honour in which he is held.

DR. H. E. HAMBERG has communicated to the Swedish Academy of Sciences a paper on the radiation of the upper clouds round barometric minima, prepared from the cloud observations available at Upsala Observatory for the years 1874-89. The arrangement of the highest clouds-cirrus and cirro-stratus-in the form of parallel bands has long been noticed by meteorologists in this country, and various papers on the subject have been written by Mr. W. C. Ley, MM. Hildebrandsson, Köppen, and others; and the movements of these clouds, in conjunction with the wind prevailing at the earth's surface, are at times sufficient to determine approximately the direction in which an atmospheric disturbance exists, even with out the use of synoptic charts. For instance, a barometric minimum often exists in a direction nearly perpendicular to that of the radiation, and, generally, on that part of the horizon where the bands of upper clouds are most dense, or whence they seem to radiate, but it is always necessary to take into account the direction of the wind at the earth's surface. The author draws the following conclusions from his investigation :-(1) The radiation of the upper clouds is closely connected with barometric minima. (2) Near a barometric minimum, with the pressure below 299 inches, the radiation forms with the radius of the depression an angle of about 70°, the deviation of the radiation from the direction of the surface wind being positive (i.e. to the right), by some degrees, on the south-west of the barometric minimum, and negative on the south-east of it. (3) Further from the barometric minimum, with a pressure of 29.9 inches and above, the inclination to the radius is rather greater, about 75°, except where the barometric minimum lies to the north of the place of observation, in which case it is much lower. (4) The angle formed by the radiation is generally greater in the rear of a barometric minimum, reaching nearly 90° in a high pressure; on

the other hand, it is smaller in front, especially to the south-east of a minimum, and further from the centre. (5) Compared to the general circulation of the air in a barometric minimum, the radiation of the upper clouds most resembles the direction of the wind near the earth's surface. The meaning of this last sentence is not obvious; but the other conclusions agree, on the whole, with the views of other meteorologists who have studied the subject.

THE Meteorologische Zeitschrift for November contains a summary, by Dr. J. Hann, of the meteorological observations taken at Cairo from 1868-88. The observations have been published in extenso, together with a good introduction upon the climate, in the Bulletin of the Egyptian Institute, and although similar observations have occasionally been published before, the present series contains much new and useful material. The most striking feature in the climate of this part of Egypt is the Chamsin, the hot and dust-bearing wind which makes its appearance in March or April for about three to four days at a time, and robs a large portion of the trees of their leaves. In the intervals during which this wind is not blowing the weather is pleasant and clear during spring-time, and the nights fresh and calm. During the summer the north winds prevail, with high temperature, very clear air, and great dryness. Towards September humidity appears with the rise of the Nile, the ground is at times covered with heavy dew, and the heat becomes oppressive on account of the moisture. In October and November fog occasionally occurs in the morning, and rain begins to fall. After this season the temperature is uniform

and pleasant. Snow is unknown, frost very seldom occurs, and rain is not very frequent. The absolute maximum temperature of the 21 years period was 117° in August 1881, which was also closely approached in May 1880, viz. 116°4. The absolute minimum was 28°4 in February 1880, and the mean annual temperature was 70°5. Rainfall is only given for the years 1887-88, in which o 87 and 167 inches fell respectively. The relative humidity sinks at times even on a daily average to 12 per cent., and has been known to fall as low as 3 per cent. at certain hours. Thunderstorms and hail are very rare. The original work contains a long investigation on the connection between the height of the Nile and the weather, a comparison between the present climate and that at the beginning of this century, and several carefully prepared diagrams referring to all meteorological elements.

THE refraction and velocity of sound in porous bodies allowing passage of sound, such as sponge, wadding, felt, &c., have been recently made a subject of investigation by Herr Hesehus (Rep. der Physik). His plan was to make a plane convex spherical lens of iron wire net, and fit it, filled with the porous body (variously condensed), into an aperture in a screen. A pipe of

variable tone was sounded on one side, and the behaviour of a sensitive flame noted on the other. From the distance of the focus, when found, could be deduced the refractive index and the velocity in the lens. The refrangibility grows with increasing density of the porous substance, while the velocity on the average is lessened; the latter is also less, the greater the sound-wave. The author gives details of experiments in which the velocity varied from 146 to 261 metres per second (with ebonite shavings). From an empiric formula which he gives, he makes deductions regarding the propagation of sound in tubes, considered as only a special case of its spread through the pores and passages of a porous body. He hopes further research in this field may do something to elucidate the passage of light and electricity through media.

THE Report of the United States Commissioner of Education has been submitted to the Secretary of the Interior. It says that the usefulness of the Bureau depends directly upon what it

prints and publishes, and therefore urges an appropriation of 30,000 dollars for general printing for the fiscal year 1892-93, and makes a special request for a specific appropriation of 20,000 dollars to continue the series of educational histories of the several States. The Commissioner reports that there were enrolled in 1889-90 in the public schools of the United States of elementary and secondary grade 12,686,973 pupils, as against 9,867,505 in 1880. The enrolment formed 20:27 per cent. of the population of 1890. The average daily attendance of pupils on each school day in 1890 was 8,144,938. The whole number of public school teachers in the past year was-males, 125,602; females, 233,333. The total amount expended during the past fiscal year for public school purposes was 140,277,484 dollars, is against 63,396,666 dollars in 1870, and 78,004,687 dollars in 1880. The expenditure per capita of population in 1880 was 1.56 dollars, while in 1890 it was 2.24 dollars.

THE U.S. Bureau of Education has issued, as one of its "Circulars of Information," an excellent paper on Sanitary Conditions for School-houses," by Albert P. Marble, Superintendent of the Public Schools of Worcester, Mass. Dr. Marble has for many years studied the problems of ventilation, heating, lighting, draining, and school-house construction; and his suggestions are well worthy of consideration in this country, as well as in America. The value of the Circular is increased by an appendix, in which are given a number of designs of school buildings of various sizes, carefully selected with a view to commodiousness, healthfulness, and economy of construction. In an official statement prefixed to the Circular, attention is especially called to a series of nineteen plates constituting the prize designs selected and published by the State of New York in 1888.

IN the interesting paper on insectivorous plants, read before the Royal Horticultural Society on September 22, 1891, and now published in the Society's Journal, Mr. R. Lindsay refers to the experiments by which Mr. Francis Darwin has shown the amount of benefit accruing to insectivorous plants from nitrogenous food. Mr. Lindsay says his own experience in the culture of Dionæa is that when two sets of plants are grown side by side under the same conditions in every respect, except that insects are excluded from the one and admitted to the other, the latter, or fed plants, are found to be stronger and far superior to the former during the following season. He points out the importance of remembering that the natural conditions under which these plants are found are different from what they are under cultivation. In their native habitats they grow in very poor soil and make feeble roots, and under these conditions may require to capture more insects by their leaves to make up for their root deficiency. Under culture, however, fairly good roots for the size of plant are developed. “Darwin," says Mr. Lindsay, "mentions that the roots of Dionæa are very small: those of a moderately fine plant which he examined consisted of two branches, about one inch in length, springing from a bulbous enlargement. I have frequently found Dionæa roots six inches in length; but they are deciduous, and I can only conjecture that the roots mentioned by Darwin were not fully grown at the time they were measured. What is here stated of the natural habits of Dionæa applies more or less to all insectivorous plants."

GOOSEBERRIES are so much liked by most people that it is very desirable the season for them should, if possible, be prolonged. According to Mr. D. Thomson, who has a good paper on gooseberries in the current number of the Journal of the Royal Horticultural Society, this can be done easily in the northern part of Great Britain. At Scotch shows it is quite usual to see fine fresh gooseberries about the middle of September. These, as a rule, are gathered from ordinary bushes that have perhaps

been shaded with mats or canvas after becoming ripe. The best way to lengthen out the season of gooseberries, Mr. Thomson says, is to plant a portion of a wall with a due north aspect with some Warringtons, and train them on the multiple-cordon system, and keep the laterals spurred in precisely the same way as is adopted with red currants on fences or walls, or in fact with gooseberry bushes grown in the ordinary way. The main shoots should not be closer than 10 inches. If a coping of wood be placed on the, wall to throw off wet, a net being used to protect the fruit from birds, the gooseberries can be kept fresh till far into October, and are then very useful and acceptable.

THE authorities responsible for the working of the free public libraries of Manchester cannot complain that these institutions are inadequately appreciated. From the Thirty-ninth Annual Report on the subject to the Council of the city we learn that, during the year ended September 5, 1891, the number of visits made by readers and borrowers to the Manchester libraries and reading rooms reached an aggregate of 4,327,038, against a total for the preceding twelve months of 4, 195, 109. The number of volumes lent for home reading was 702,803. Of these, only thirteen are missing.

AT a recent meeting of the Chemical Section of the Franklin Institute, Dr. Bruno Terne read a paper on the utilization of the by-products of the coke industry, In the course of his remarks he said it seemed strange, and nevertheless was a fact, that, with all the ingenuity of the American people in the advancement of the purely mechanical part of the technical industries, they have been and are still slow in the development of chemical industries. "If," said Dr. Terne, "you will visit our coal region to-day, you will find the nightly sky illumined from the fires of the coke ovens, and every one of the brilliant fires bears testimony that we are wasting the richness of our land in order to pay the wiser European coke manufacturer, who saves his ammonia and sends it to us in the form of sulphate of ammonia; and who also saves his tar, which, after passing through the complex processes of modern organic chemistry, reaches our shores in the form of aniline dies, saccharin, nitrobenzol, &c." Dr. Terne thinks that every pound of ammonia used in America ought to be produced there, and that every pound of soda should be made from American salt wells by the ammonia process.

MR. COLEMAN SELLERS contributes to the December number of the Engineering Magazine, New York, the first of a series of articles on what he calls "American Supremacy in Mechanics." Incidentally, he notes that most English inventions brought to the United States have to be "Americanized, simplified, made accessible in the case of machinery, and constructed with a view to ease of repair as well as to durability when under the care of careless attendants." Mr. Sellers does not think it would be worth the while of Americans to copy "the solidity and immense weight that some deem a merit in English machinery."

ACCORDING to the "World's Fair Notes," sent to us from has been making excavations in the mounds of Ohio, made an Chicago, the party which, under the direction of Mr. Putnani, important discovery on November 14. While at work on a mound 500 feet long, 200 feet wide, and 28 feet high, the excavators found near the centre of the mound, at a depth of 14 feet, the massive skeleton of a man incased in copper armour. The head was covered by an oval-shaped copper cap; the jaws had copper mouldings; the arms were dressed in copper, while copper plates covered the chest and stomach, and on each side of the head, on protruding sticks, were wooden antlers ornamented with copper. The mouth was stuffed with genuine pearls of immense size, but much decayed. Around the neck was a necklace of bears' teeth, set with pearls. At the side of this skeleton was a female skeleton.