Such records have been made for more than thirty years in France and Germany, and surely we must have here, if any. where, a sufficient proof of a forest's influence.

Admitting that we have perfect instruments and careful observers, there still remains a most serious doubt as to the immediate environment of each gauge and as to the possibility of a direct comparison. It is probable that no two gauges 2000 feet apart can be placed so as to catch the same amount of rain, though to all appearances the exposure is faultless in each

case.

Extreme caution is therefore needed in arriving at conclusions from comparisons between gauges in forests and in the open. One of the best of all researches in this line has been conducted at Nancy, in France. Within a distance of five or six miles there have been four stations established. At Nancy in the open, and at Belle-Fontaine in the forest; and, 500 feet higher vertically, Amance (open) and Cinq-Tranchées (forest). At Nancy and Belle-Fontaine the observations extend over twentyfive years. A comparison of the records in groups of eight, eight, and nine years was made, with the result that while the first eight years showed a very slight excess in the forest rainfall over that in the open field, in the last nine years (including 1894, last published) the open station showed a little more rain than the forest station. These observations were made with particular care, for the purpose of exactly determining the influence, and may be relied on if the environments of the gauges were comparable. At Amance (open) and Cinq-Tranchées (forest) the observations have not been quite so regular, though there are twenty-five full years of records at these two stations, but not the same years as at the other stations. The comparison in this case makes the rainfall more than 20 per cent. greater in the forest than in the open. It should be borne in mind, however, that these two stations are on an eminence, and are not strictly comparable, and this result cannot vitiate that at the two other stations, which shows no effect.

In Germany we have a rather remarkable record of a slightly different character. Lintzel is a station on the Luneburg Heath, which began to be planted with trees in 1887, at the rate of 1000 to 1500 acres a year, and in a few years over 8000 acres were covered. In the midst of this forest is the meteorologic station in an open field of some seventy-five acres. Before planting the forest, 97 per cent. of the surface was field, meadow, or heath, and afterwards 80 per cent. was forest and 20 per cent. was roads, open field, and heath. Around this station, pretty evenly distributed, and within fifty miles, there are thirteen rainfall stations which have been carefully established, and presumably are comparable with the Lintzel station in the midst of the growing forest. There are no means of knowing whether any of these stations have been changed or not, but for our purpose we may consider the material homogeneous, and treat it accordingly. Records from 1882 to 1896 (fifteen years) are available. Charts were prepared for each year showing the ratio between the Lintzel record and that at each station of the thirteen. The results do not show that the afforestation has had any appreciable effect upon the precipitation; in 1884 the ratio was 101, while in 1893, nine years later, it was 96. It is probable, however, that no definite and unassailable result can ever be obtained either by the method adopted in France or this later one in Germany. The rainfall is so variable within a distance of even a mile or two; and it is so difficult, if not impossible, to obtain similar environments at all the stations, that no decisive result can be obtained. It will be readily seen that the multiplication of stations will do no good, and, above all, that the observation of rainfall under trees in a forest is absolutely useless for any such discussion or study as this.

Need of Further Evidence.

It seems probable that if two or three lines of stations could be established a mile or two apart on four sides of an enormous forest, each line to have a dozen stations or so, about 3000 feet apart, four of the stations to be outside of the forest, and the others each in a large cleared space of at least two acres extent in the forest, something decisive might be obtained. It should be noted, however, that from the evidence already accumulated there would be very little to be gained by a further study of the question. It is certain that the effect, if there be one, is almost inappreciable. The favouring conditions over the forest are balanced by those not favouring, and the integrated effect is practically the same in the two cases.

Prof. H. F. Blanford determined from a most careful

series of records, from which all known errors had been eliminated, that the forest had a tendency to give 2 per cent. more rain than contiguous open fields. That is, if an open place had 50 inches of rain in a year, a near-by forest would have only 51 inches, which is practically inappreciable.

It would be an interesting study to select all those cases in experiments in forest and near-by fields in which the wind was blowing either from the forest to the field, or vice versa. It is evident that if there is any effect on rainfall by the forest, it would be vitiated, if not exactly reversed by such winds.

There is a class of visual observations which seem to show an effect upon rainfall by the forest. Probably many have seen heavy clouds passing over a plain, but which only precipitated as they passed over a forest. Also in a hilly region it is a frequent phenomenon that fog and low-lying cloud hover near a forest, and not over an open plain. One also notes very often, in passing into a forest on a damp day, that the trees drip moisture, possibly condensed from moisture evaporated from the damp earth underneath. Observations of this nature, however, cannot ordinarily be checked by instrumental means, but show in a general way that the forest tends to conserve vapour and moisture which in the case of the open field would be diffused into the atmosphere.

UNIVERSITY AND EDUCATIONAL
INTELLIGENCE.

DR. G. H. RENDALL, the Principal of University College, Liverpool, has been appointed Head-Master of Charterhouse School.

SIR JOHN GORST, in the course of an address at Bristol on Thursday last, is reported by the Times to have said that the promotion of technical education was confronted by two obstaclesthe backward condition of elementary education and the want of organisation in the provision of secondary education. A good sound system of elementary education must be the groundwork for higher education, and he urged reform of the system which at present relieved children from compulsory attendance when inadequately equipped. The improvement of the organisation of secondary schools was really a matter for the people themselves. There was nothing to prevent technical instruction committees from becoming thoroughly representative and effective organisations.

THE most satisfactory point to us in the Report just issued by the Oxford University Extension Delegacy refers to the Extension College at Reading. The college is doing excellent work, more particularly in agriculture, and has amply justified its existence. New buildings are, however, imperatively needed, and in response to an appeal for 12,000/., 9,000/. has already been promised, and the new wing has been begun. The building scheme, planned four years ago, will be completed by next summer, and H. R. H. the Prince of Wales has promised to perform the opening ceremony. The educational work of the college has been attended with great success during the past With regard to the courses of lectures delivered under the auspices of the Delegacy during the year 1896-97, we notice that out of a total of 146 courses, only nineteen were on scientific subjects.

year.

IN the course of a presidential address recently delivered before the Kansas Academy of Sciences, Prof. S. W. Williston severely criticised the system of education which makes language studies compulsory, and all, or nearly all, the sciences optional. Many educationists will find themselves in agreement with the following opinions expressed by Prof. Williston :- -"I claim broadly and emphatically that the natural sciences, any or all of them, are as valuable and as necessary as pure cultural studies as are the languages; that intelligent and successful study of them will do as much, if not more, in making the student a broad man, a successful man, as will the study of Latin or Greek. And they will do more in making him an honest man. Nowhere in all the broad field of knowledge will he learn better to think exactly than in the natural sciences. Now here will he be more impressed with the importance of truth for truth's sake. . . . Were I, then, to say what the universities and colleges ought to do, it would be this: make all the ancient language requirements for admission optional, and demand as much preparation in the physical and biological sciences as in the foreign languages. The preparation in English should be made far more rigorous

and thorough. In the college course, if anything besides English is required, and I think there should be, I would have the natural science as necessary a part of the education as language and mathematics. I would not have it possible for a student to graduate from the college without having studied, and thoroughly studied, mathematics as far as trigonometry, at least one foreign language, and at least one physical and one biological science. And I do not mean a few weeks of study in any of these branches, but exhaustive, careful, critical study. The methods of study in all these branches are diverse, and are absolutely essential for symmetrical mind-building,"

SOCIETIES AND ACADEMIES.
LONDON.

Royal Society, November 25.-"Further Note on the Transplantation and Growth of Mammalian Ova within a Uterine Foster-Mother." By Walter Heape, M.A., Trinity College. Cambridge.

In 1890 an experiment was recorded (Roy. Soc. Proc., vol. xlviii.), designed to show that it is possible to make use of the uterus of one variety of rabbit as a medium for the growth and complete foetal development of fertilised ova of another variety of rabbit. The experiment was further undertaken in order to determine what effect, if any, a uterine foster-mother would have upon her foster-children, and whether or not the presence, during development, of foreign ova in the uterus of a mother would affect offspring of that mother present in the uterus at the same time. In this experiment, two fertilised ova obtained from an Angora doe rabbit which had been inseminated thirty-two hours previously by an Angora buck, were inserted into the fallopian tube of a Belgian Hare doe, which had been inseminated three hours before by a buck of the same breed as herself; and in due course the Belgian Hare doe littered six young, four of which were Belgian Hares, while the other two were Angoras. This year experiments were made with Dutch and Belgian Hare rabbits, and the method adopted was the same as that described above, the result being that the Belgian Hare foster-mother gave birth to seven young, of which five were Belgian Hares and two were apparently Dutch. Both these Dutch young were, however, irregularly marked, and it appeared possible, after all, either (1) that the Belgian Hare fostermother had influenced the Dutch fertilised ova, or (2) that these two young were really a cross between Dutch and Belgian Hare.

In order to test the first of these possibilities, the same Dutch buck was put to a tried, thoroughbred Dutch doe, and she produced a litter, every one of which was badly marked, thus showing that the bad marking of the foster-children can be justly attributed to their father's influence. The second possibility was more difficult to test. A cross between the Dutch buck and the Belgian Hare foster-mother was obviously possible, for when the foreign Dutch segmenting ova were introduced into the fallopian tube of the Belgian Hare foster-mother, they were still surrounded by spermatozoa from the Dutch buck, which were still alive, though failing in vigour. But the Belgian Hare doe had been inseminated by a Belgian Hare buck just before the operation, and the spermatozoa from this buck would arrive at the end of the fallopian tube before ovulation took place; it would be at least twenty-four hours younger than the foreign Dutch spermatozoa, and both more vigorous and in far greater numbers than the latter. The possibilities are distinctly in favour of the host of younger and more vigorous Belgian Hare spermatozoa beating the few older and less vigorous, foreign, Dutch spermatozoa in the struggle for the Belgian Hare ova; but, at the same time, it is possible that the latter won. The only way to test this at all seemed to be by crossing the same Dutch buck with Belgian Hare does, and comparing the offspring of such crosses with the young foster children. This was done, and two Belgian Hare does each produced, in consequence, five young. Of these, three were Belgian hares splashed with white, one was black and white, three were fawn or fawn and white (the fawn being mixed with a delicate bluish dun shade), and three were thoroughbred Belgian Hares. The father's influence was seen in the introduction of white and in the fawn and dun colours. None of the young, however, at all closely resembled the Dutch breed.

With regard to the foster-children, one of them died at an

early age, but the second lived, and is now more typically Dutch than it was when younger; it is coloured and shaped remarkably like the Dutch doe from which the foreign fertilised ova were obtained. The remarkable likeness is in itself very strong evidence of the origin of this young one, and when considered in conjunction with the results obtained by crossing the Dutch buck with Belgian Hare does, there can be little doubt it was derived from Dutch parents. This result, supported by the result obtained in 1890, is greatly in favour of the contention, that it is possible to make use of a uterine foster-mother, and to do so without thereby influencing any of the young which are nourished by her.

It is worthy of notice, if the above is true, that in case telegony be actually demonstrated, the characteristics of a primary husband transmitted to the offspring got by a secondary husband, can only be so transmitted through the ova of the mother.

"Mathematical Contributions to the Theory of Evolution. IV. On the probable Errors of Frequency Constants and on the Influence of Random Selection on Variation and Corrlation." By Karl Pearson, F.R.S., and L. N. G. Filon, University College, London.

A brief indication of the nature of the contents of this paper is given on p. 210.

December 9-"On the Calculation of the Coefficient of Mutual Induction of a Circle and a Coaxial Helix, and of the Electromagnetic Force between a Helical Current and a Uniform Coaxial Circular Cylindrical Current Sheet." By J. Viriamu Jones, F.R.S.

Zoological Society, December 14.-Lieut.-Colonel H. H. Godwin-Austen, F. R. S., Vice-President, in the chair. - Mr. G. A. Boulenger, F.R.S., offered some further remarks upon the Siluroid Fish, Vandellia cirrhosa.-A communication was read from Dr. E. A. Goeldi, “On Lepidosiren paradoxa from the Amazons." This memoir treated of the geographical distribution of the Lepidosiren on the Amazons, and of its external structure and dimensions, and gave an account of its habits in a natural and captive state.-Mr. J. Graham Kerr gave an account of his recent expedition, along with Mr. Budgett, to the Chaco of Paraguay in quest of Lepidosiren: and made remarks on its habits as there observed. Mr. Kerr also gave a general account of the early stages of its development, drawing special attention to the presence in the larva of external gills and a sucker similar to those of the Amphibia.-A communication was read from Dr. A. G. Butler, containing a list of thirty-three species of butterflies obtained by Mr. F. Gillett in Somaliland during the present year, and giving the dates of the capture of the specimens and their localities.-Mr. Oldfield Thomas read a paper entitled "On the Mammals obtained by Mr. A. Whyte in North Nyasaland, and presented to the British Museum by Sir H. H. Johnston, K. C.B.; being a fifth contribution to the Mammalogy of Nyasaland." This memoir contained notes on sixty-one species of Mammals, four of which were characterised as new, viz. Macroscelides brachyrhynchus malos, Crocidura lixa, Myosorex soulla, and Graphiurus johnstoni.-A communication was read from the Rev. O. Pickard Cambridge, F. R.S., describing a new genus and species of Acaridea (Eatonia scopulifera) from Algeria.-A communication by Mr. J. Stanley Gardiner, "On some collections of corals of the family Pocilloporide from the South-west Pacific Ocean," was read by the author. Twenty species of the genus Pocillopora and one of the genus Seriatopora were enumerated and remarked upon, five species of the former genus being described as new, viz. Pocillopora septata, P. obtusata, P. coronata, P. rugosa, and P. glomerata.-Mr. W. E. de Winton gave an account of a collection of Mammals from Morocco, made by Mr. E. Dodson on behalf of Mr. J. I. S. Whitaker. Twenty-one species were enumerated as represented in the collection, of which the following were described as new: Crocidura whitakeri, Mus peregrinus, and Lepus atlanticus.

DUBLIN.

Royal Dublin Society, November 17.-Dr. F. T. Trouton, F. R.S., in the chair.-Dr. G. Johnstone Stoney, F. R.S., presented a paper upon atmospheres upon planets and satellites (see p. 207). Mr. W. E. Wilson, F.R.S., read a paper upon the apparent cometary nature of the spiral nebula in Canes Venatici. paper was illustrated by a remarkably fine photograph of the nebula taken in February 1897, by the author.-Dr. F. T. Trouton read a paper upon the arrangement of the crystals of

The

certain substances on solidification.-Prof. A. C. Haddon presented a paper upon the Actiniaria of Torres Straits. This account of the Actiniaria is based mainly on the collections made by the author in 1888-9, supplemented by descriptions published by Mr. Saville-Kent in his works 'The Great Barrier Reef of Australia" and "The Naturalist in Australia." In order to render the paper more complete, allusions are made in it to genera which are not recorded, from Torres Straits. In a second paper, Prof. Haddon described a new species of Actiniaria from Oceania-Phellia Sollasi This was collected by Prof. Sollas in the lagoon at Funafuti, Ellice Group, W. Pacific, in 1896. -The following objects were exhibited at this meeting The Coccoliths of Dublin Bay, by Mr. H. H. Dixon, and Prof. J. Joly, F.R.S.-A collection of economic plant products from the Gold Coast, by Prof. T. Johnson.

ST. LOUIS.

Academy of Science, December 6.-Mr. Julius Hurter exhibited specimens of a considerable number of reptiles and batrachians, mostly of southern origin, which had been collected by him during the past season, and were additions to the known fauna of Missouri. Among the more interesting additions were the cotton-mouth moccasin, the banded water snake, Holbrook's water snake, the little brown snake, the Louisiana mud turtle, the chestnut-backed salamander (first detected west of the Mississippi River by Mr. Colton Russell), and the marbled salamander.Mr. II. von Schrenk exhibited a series of specimens and drawings illustrating some of the injuries inflicted on the trees of St. Louis by the tornado of May 1896, showing not only the formation of double twig elongation and growth rings, but the exfoliation of the bark and the consequent drying out of 50 per cent. or more of the wood through the trunk and branches, in several species.

NEW SOUTH WALES.

Linnean Society, October 27.-Prof. J. T. Wilson, President, in the chair.-Descriptions of new species of Australian Coleoptera, Part 4, by Arthur M. Lea. Thirty-four species, principally belonging to the Curculionida, were described as new ; with critical notes and remarks on synonymy.-On the lizards of the Chillagoe district, North Queensland, by Dr. R. Broom. Twenty-three species were collected during a six months' residence at Muldiva, seventy miles west of Herberton, a district in which during eight months of the year (April-December) as a rule there is practically no rain. A species of Lygosoma was described as new. -On a Trachypterus from New South Wales, by J. Douglas Ogilby. In this paper the author gave a detailed description of a young example washed ashore near Newcastle, and reviewed at length our present knowledge of the genus in the south-western Pacific.-Contributions to a more exact knowledge of the geographical distribution of Australian Batrachia, No. 5, by J. J. Fletcher. The present contribution is based upon the examination of collections from Tasmania and West Australia. In the British Museum Catalogue (second edition) seven (? eight) species are attributed to Tasmania, and fourteen to West Australia. Three additional species are now recorded for the former Colony, and six for the latter, including an undescribed species of Crinia belonging to the group having the abdominal surface non-granulate.-Mr. Froggatt exhibited a number of scale insects (Eriococcus coriaceus, Mask.), upon a twig of Eucalyptus, among which had been placed a great number of the eggs of the scale eating moth Thalpochares coccophaga, Meyr. The eggs are pale pink, circular, and beautifully ribbed The scales were infested with the larvae of Cryptolemus montrouzieri, Muls., a useful small black ladybird beetle. Both these enemies of Eriococcus are of great economic value, as the moth larvæ have now taken to eating the olive scale (Lecanium olea, Sign.), and the ladybird beetle is bred both in New Zealand and America. Also living specimens of our largest white ant, Calotermes longiceps, Froggatt, which were taken out of a log of fire-wood, and had already been in captivity for over two months.

TUESDAY, JANUARY 4.

ROYAL INSTITUTION, at 3.-The Principles of the Electric Telegraph:
Prof. Oliver Lodge, F.R.S.
ROYAL VICTORIA HALL, at 8.30.-Coal: W. F. Rudler.
WEDNESDAY, JANUARY 5.

GEOLOGICAL SOCIETY, at 8.-On the Structure of the Davos Valley: A.
Vaughan Jennings.-Sections along the Lancashire, Derbyshire, and
East Coast Railway, between Lincoln and Chesterfield: C. Fox-Strang-
THURSDAY, JANUARY 6.
ROYAL INSTITUTION, at 3.-The Principles of the Electric Telegraph:
Prof. Oliver Lodge, F.R.S.

ways.

FRIDAY, JANUARY 7. GEOLOGISTS' ASSOCIATION, at 8.-A Brief Account of the Excursions in the Urals, down the Volga, in the Caucasus, &c, made in connection with the Interntional Geological Congress held in Russia, August-September, 1897 L. L. Belinfante.

SATURDAY, JANUARY 8. ROYAL INSTITUTION, at 3.-The Principles of the Electric Telegraph: Prof. Oliver Lodge, F.R S.

BOOKS, PAMPHLETS, and SERIALS RECEIVED. BOOKS.-L'Electro-chimie A. Minet (Paris, Gauthier Villars).—Intreduction to the Study of Organic Chemistry: J. Wade (Sonnenschein) Natürliche Schöpfungs-Geschichte: Prof. E. Haeckel, 2 Vols., Neurte Umgearbeitete Auflage (Berlin, Reimer).-Notes on Carpentry and Joinery T. J. Evans, Vol. 1 (Chapman).-What is Life?: F. Hovenden (Chapman). The Collected Mathematical Papers of Arthur Cayley, Ve xiii. (Cambridge University Press).-Physikalisch-Chemische Propar dentik Prof. H. Griesbach, Zweite Hälfte, 2 Liefg (Leipzig, Engelma n PAMPHLETS.-Magnetic and Pendulum Observations: G. R. Punan (Boston, Mass.).—Hand-Guide to the Botanic Gardens, Buitenz g (Batavia, Kolff).

SERIALS.-Traité Encyclopédique de Photographie: Dr. C. Fabre Cinq Fasc. B. (Paris, Gauthier Villars).-Journal of the Royal Micro(Macmillan) -Quarterly Journal of Microscopical Science, December scopical Society, December (Williams).-Century Magazine, January (Churchill).-Natural Science, January (Dent).

THURSDAY, JANUARY 6, 1898.

CAYLEY'S PAPERS.

The Collected Mathematical Papers of Arthur Cayley, Sc.D., F.R.S. Vols. viii., ix. Pp. liv + 570, xvi + 622. (Cambridge at the University Press, 1895, 1896.)

THES

HESE two volumes form the first of those published after Cayley's death in 1895. The first thirtyeight sheets of Vol. viii. were revised by the author, who added a note on one paper (No. 518); the duty of editing the rest of the papers was entrusted to Prof. Forsyth, who has very faithfully carried out the plan and arrangements which, in the absence of definite instructions, he was able to infer from the previous volumes.

Perhaps the reader's first impression after surveying these 144 papers, mostly published in the years 1871-77, is that they are very miscellaneous, and that comparatively few are of paramount importance. The fact is that Cayley is, as it were, brought into unfavourable comparison with himself; short notes on special problems of geometry and analysis, and solutions of Smith's Prize papers cannot rank with the immortal "Memoirs on Quantics," or some of the earlier geometrical papers, such as that upon plane cubic curves. But it is unreasonable to expect an artist to produce an uninterrupted succession of masterpieces; and it is to be remembered that Cayley seldom, if ever, wrote upon any subject without developing some instructive point or giving an example of his own characteristic elegance.

In trying to give some account of the more important of these memoirs it will be convenient to take the geometry and the analysis separately. Not that the boundary line is very easy to fix: Cayley was never a geometrician in the sense in which the word may be applied to Apollonius or Steiner. But some of the papers have an interest mainly geometrical, although the methods used are almost wholly algebraic; and with them we will begin.

Perhaps the most important are those which deal with transformation, correspondence, and the singularities of algebraical curves and surfaces. With these difficult theories Cayley dealt in a masterly way he avoided, as if by instinct, the many opportunities of mistake which present themselves in a method which is largely enumerative, and he had the gift of predicting general results from the consideration of special cases.

Coming next to what may be called the metrical geometry of surfaces, which has developed so greatly in recent years, we have papers on curves of curvature, on geodesics on quadrics, and on orthogonal surfaces. To this group may perhaps be added a paper on evolutes and parallel curves, though this is rather meant to illustrate the non-Euclidian geometry.

There are three monographs, on Steiner's surface, on the centro-surface of an ellipsoid, and on the configuration of the twenty-seven lines of a cubic surface, which are in various ways highly characteristic. As models of analytical skill they are admirable; and as helps to the understanding of the geometrical figures with which they deal, they are of great service. But it is curious to see how chary the author is in giving illus

217

trative diagrams. There are, indeed, two figures in the paper on the surface of centres; but why, we ask, did Cayley not give a series of contour lines of the surface? or again, with still more reason, in the case of Steiner's surface? Then the paper on the twenty-seven lines of a cubic surface is so quaint in its topsy-turveydom as almost to suggest Mr. W. S. Gilbert as joint author. Here we have a projective configuration which may be realised with the help of a bundle of sticks and without any measurement whatever. What Cayley did was to take a model by Dr. Wiener, measure approximately the coordinates of a number of points upon it, thence find the approximate equations of the lines, and finally adjust the equations so as to satisfy the geometrical conditions! Of course there is reason in this seeming perversity : by the projective method it is not easy to get a convenient arrangement of the sticks, whereas Cayley's equations make it possible to construct a string model on a cardboard frame without a tiresome series of preliminary experiments.

The poristic polygons of Poncelet appear to have had for Cayley a perennial charm: we have here two papers suggested by Poncelet's results; one "On the porism of the in-and-circumscribed polygon, &c.," which treats of the original problem, and the other "On the problem of the in-and-circumscribed triangle," which really deals with a rather different and more general theory. Cayley, like many others, does not seem to have been aware (at least in 1871) that the complete algebraical solution of the Poncelet problem was published in 1863 in a paper by M. Moutard, which formed part of the appendix to Poncelet's Applications d'Analyse à la Géométrie." Not only is this so, but, as Halphen pointed out, this paper contains the first fully satisfactory treatment of the multiplication of the argument in elliptic functions.

Before passing on from the geometrical papers, attention should be called to the very interesting series of notes on the mechanical description of curves. This is

a promising field of research, and the results could hardly fail to be of interest, especially to those who like to see the deductions of theory embodied in an actual geometrical figure. There is an æsthetic satisfaction in this contemplation: and, moreover, a really correct figure often suggests geometrical truths that would otherwise be overlooked.

Of the analytical papers the one which has been most appreciated in this country is, beyond question, the short paper" On the theory of the singular solutions of differential equations of the first order" (Messenger, vol. ii. (1873) pp. 6-12). Here Cayley's power of giving to analysis a geometrical interpretation appears to the best advantage. If we have an algebraical relation f(x, y, p) = o in which enters to the degree s, then this associates with any point (x, y) a series of s (real or imaginary) directions corresponding to the different values of p in other words, the differential equation really expresses that the plane of reference is covered with 2 tiny s-rayed stars. The primitive (x, y, c) =0 gives a family of ∞1 curves each made up of ∞1 selected rays. Now if we eliminate p from_ƒ(x, y, p) = 0, aƒ apo, we obtain a locus of points (x, y) at each of which two rays coincide in direction; where this happens either two consecutive curves (x, y, c) = o touch, ox

:

two non-consecutive curves touch, or (x, y) is a cusp or point of self-contact of one particular curve (x, y, c)=0. Thus we may have the envelope of the family of curves, a tac-locus, or a locus of cusps or of points of selfcontact. On the other hand if we eliminate c from (x, y, c) = 0 and əpəc = o, we get the locus of intersection of consecutive curves : this may include besides the envelope proper, a locus of nodes, of cusps, or of multiple points of higher order (as, for instance, points of self-contact or triple points). The only outstanding difficulty is the degree of multiplicity in which the singular loci, distinct from the envelope, are involved in the two discriminants.

There are six papers on the transformation of elliptic functions, the most important being No. 578. This contains an exposition of the Jacobian theory, Sohnke's modular equations with additions, and a discussion of the singularities of some of the modular curves. It is remarkable that Cayley, like Kronecker, adhered firmly to Jacobian methods, and never seems to have worked with the Weierstrassian forms. Perhaps just now there is a rather exaggerated tendency in the other direction: as Prof. Klein has pointed out, both theories are selfconsistent and form, in a sense, the first and second stages in a complete discussion of periodic functions.

There is not very much about invariants and covariants; No. 525 is an interesting example of a quadratic transformation, and the papers on "trees," although ostensibly intended for application to chemistry, were suggested by the invariant calculus.

In arithmetic there is a table of reduced binary cubics with their Hessians, which is a development of Arndt's results. Cayley gives the composition tables for the Hessians.

Volume ix. contains eleven papers dealing more or less with astronomy and dynamics; and it may be worth while to notice that this volume also contains a reprint of the British Association "Report on Mathematical Tables."

Many interesting special points suggest themselves to the reader thus, to mention only three, very different in character, the very simple and pretty proof of Vandermonde's theorem (viii. p. 465) might very well find a place in an elementary text-book of algebra; we are told (ibid., p. 188) how a theoretical error was detected by a numerical calculation; and (ibid., p. 397) there is an unverified conjecture that every surface of negative deficiency may be derived by a rational transformation from a cone whose deficiency is equal to that of the surface with its sign changed. G. B. M.

colleges"; and he proceeds to point out that if physics is to possess much disciplinary value, it must be taught by laboratory methods. Experimental work thus finds a prominent place in his book, which may, in fact, be roughly described as a series of experiments, mostly suitable for repetition by young students, connected by short discussions of a theoretical character.

With the author's object we imagine that most teachers of physics will cordially sympathise. That experiment is the means whereby a knowledge of physics should be acquired by beginners, is as clear now-a-days as it is that the means itself is open to improvement—at any rate, in its early stages. Whether the author has made the most of his opportunity is, however, less certain. Much of his work is excellent: the experiments are, for the most part, well chosen and clearly described; but after a careful perusal of his book, one's prevailing impression is that he has attempted to include too much.

A book of this kind is, of course, largely taken up with description of experimental procedure; but the space is often further occupied with matter which might, in our opinion, be left until a later stage in the student's career. Such questions as X-rays, tests for and theory of colourblindness, interference and polarisation of light, are too large for more than the briefest notice, and might therefore just as well have been omitted altogether; especially when, to mention one instance out of many, curved mirrors are dismissed with a far too scanty discussion, and no special experimental illustrations at all. It would, in our opinion, have been better to develop further the experimental treatment of the simpler parts of physics at the expense of these more elaborate phenomena. It is only in places, however, that the work is affected by this fault; and the same may be said of an occasional laxness of expression which will probably lead to mistakes on the part of young readers where it occurs. Taken as a whole, the book forms a useful addition to the elementary text-books on practical physics.

We have noticed a few points that rather need alteration. In the figure of the apparatus for determining the heat of vapourisation of water (p. 172), the long tube connecting flask and calorimeter should be provided with a trap for the steam condensed in it. The statement in italics on p. 213, that “various bodies can be brought by friction (i.e. by doing work upon them) into a condition such that they attract and are attracted," is rather misleading. It is, of course, the work done in pulling the rubber and rubbed object apart which should be emphasised. On p. 337, in the figure illustrating the motions of the air in sound waves, the arrows want altering; on pp. 308 and 310, misprints of iron for ion, and ammonium for ammonia, respectively, occur; and on p. 99, in the last column the decimal point has gone astray.

The general get-up of the book is, as one would expect, excellent; and the diagrams, which are mostly by Mrs. Nichols, are very clear and well executed. We may add that the work is almost wholly non-mathematical.

The second of the two books named at the head of this notice, forms the first part of vol. iii. of the "Elementary Practical Physics" series begun in 1885 by Prof. Balfour Stewart and Mr. W. W. Haldane Gee.

With the rapid development of the teaching of physics by laboratory methods, now in progress, has arisen the