given the subject by the municipal government for years, nothing had been accomplished, and it was, in this instance, proposed to organize a technical commission to conduct competitive tests of various methods and apparatus having for their object the suppression of smoke from boiler-furnaces. The above-named commission was accordingly formed and was assigned a credit of 8,050 francs for expenses. The commission was to select acceptable forms of furnace and report to the city government for their license and use. One hundred and ten competitors appeared, their schemes including the following:

GENERAL PLAN OF PROCEDURE.

Of the total, three-fourths were French devices, one-fifth English, 3 American, and the others of various European nationalities. A preliminary study led to the careful test of ten. These were tested to

ascertain whether they were capable of burning ordinary fuels without smoke and whether they were suitable for use in steammaking.

They were tested with rapid and with slow combustion, with operatives supplied by the makers and with firemen furnished by the commission, under the direction of first the one and then the other. The intensity of the smoke was observed and noted on a scale of five points. The usual standard methods of determining the efficiency of the apparatus were employed. The corps of observation was detailed from

ment de la Seine, Ville de Paris, République Française-Liberté, Égalité, Fraternité." n. d.

the offices of the city administration, organized and directed by the commission.

The history of legislation, as given, traces the progress of the subject in England from the time of Charles II., who, two hundred years ago, inaugurated repressive measures. In France this form of legislation began with an imperial decree in 1810. Both countries now have well-considered laws for suppression of smoke in cities. The technical history, curiously enough, begins with plans by Denis Papin. The next inventor to follow this illustrious man of science was James Watt, with his inverted draught and later arrangement of 'deadplate.' The automatic stokers,'' très usités en Amérique,' are referred to and their incidental but none the less effective, smoke reductions are described. Legislation now exists in all civilized countries, and many more or less effective devices and methods are in use for suppression of smoke.

A commission of distinguished engineers and scientific men was organized by the German government, in 1892, which, after prolonged experimental investigation, con

cluded that success had not been attained, but that the way to success was clearly indicated. This commission, in computing the heating power of combustibles from analyses, adopted the formula: 8000 C + 29000 (H―0/8) + 2500 S — 600 W; where Wis moisture.

The outcome of the work of the French

Commission was the refusal to assign a first prize, the awarding of two second prizes, of two first mentions and of one second mention. The conclusions formulated indicate that the Commission is not satisfied that a real success has been achieved, but nevertheless the researches were not without value. Like the German Commission of

1892-4, it is concluded that "The work of the Commission should be considered only as a contribution to the study of 'fumivorité,' and it is to be hoped that these re

searches may continue. There remains much to be done and a part of this collection of exhibits has very nearly attained the object proposed."

Among the specific conclusions are these:

Smoke cannot be suppressed without considerable excess of cost.

Special fuels, as anthracite, coke, fuel-gas and mineral oils, may be resorted to, and with success, where cost is not objectionable.

The chimney-top should be visible to the man at the furnace.

Prolonged trials should supplement such investigations as those prosecuted by this Commission, to ascertain the durability of the apparatus and of its efficiency.

Existing legislation, well enforced, is advised, rather than any specific new legislation.

The appendix to the report is an elaborate presentation of the logs, tables and drawings of the apparatus of the trials described in the text. The whole constitutes a very valuable contribution to the literature of the subject, in the department of applied science, and deserves to be permanently preserved in every library of applied science, beside the reports of the Franklin Institute discussion.

R. H. THURSTON.

AMERICAN MATHEMATICAL SOCIETY. THE fifth annual meeting of the American Mathematical Society was held in Fayerweather Hall of Columbia University, on Wednesday, December 28, 1898. On the two following days the Chicago Section of the Society held its fourth regular meeting in the Ryerson Physical Labratory of the University of Chicago. At the election. held at the annual meeting the following officers and members of the Council were chosen President, R. S. Woodward; First Vice-President, E. H. Moore; Second VicePresident, T. S. Fiske; Secretary, F. N. Cole; Treasurer, Harold Jacoby; Librarian, Pomeroy Ladue ; Committee of Publication, T. S. Fiske, F. N. Cole, Alexander Ziwet;

members of the Council to serve for three years, Maxime Bôcher, James Pierpont, Charlotte Angas Scott.

The Society has now completed its tenth year of continuous existence, having been organized as the New York Mathematical Society in November, 1888, and reorganized under its present title in July, 1894. The Bulletin is now in its eighth annual volume; the first number appeared in October, 1891. The present membership of the Society is 315. About ninety papers have been presented at its meetings during the past year. The Chicago Section was organized in April, 1897, and has proved from the beginning a valued addition to the Society's strength.

At the annual meeting the following papers were read:

(1) PROFESSOR M. I. PUPIN: 'On multiple reso

(3) PROFESSOR ARTHUR S. HATHAWAY: 'A new way of presenting the principles of the calculus.' (4) PROFESSOR H. MASCHKE: 'Some general theorems concerning linear substitution-groups of finite order.'

(5) PROFESSOR E. H. MOORE: 'Concerning Klein's groups of n! (n-1)-ary collineations.' (6) PROFESSOR E. H. MOORE: 'The decomposition of a modular system connected with the doubly generalized Fermat theorem.'

(7) PROFESSOR H. B. NEWSON:

Normal forms of projective transformation (second communication).'

(8) PROFESSOR H. B. NEWSON: 'A new solution of the Riemann-Helmholtz problem.'

(9) PROFESSOR H. B. NEWSON: What constitutes a continuous group?'

(10) PROFESSOR JAMES B. SHAW: 'Some quaternion integrals and their related classes of functions.'

(11) DR. H. F. STECKER: 'Non-Euclidean images of plane cubics on rotation surfaces of constant negative curvature.'

(12) PROFESSOR HENRY S. WHITE: 'Note on certain relations among fundamental covariants of a ternary cubic.'

(13) PROFESSOR J. W. A. YOUNG: 'The teaching of mathematics in the higher schools of Prussia.' F. N. COLE, Secretary.

COLUMBIA UNIVERSITY.

GENERAL MEETING OF THE AMERICAN CHEMICAL SOCIETY.

THE eighteenth general meeting of the American Chemical Society was held in New York on the 27th and 28th of December, and was in every respect a most successful and notable gathering.

The opening session was held at the rooms of the Chemists' Club, 108 West 55th Street, with an attendance of about one hundred and fifty members and visitors.

Dr. McMurtrie welcomed the visitors and then introduced Mr. Randolph Guggenheimer, President of the Council, who welcomed the Society to the city. Professor Alexander S. Webb, of the College of the City of New York, welcomed the Society to the educational and scientific institutions of the city. President C. E. Munroe re

sponded in behalf of the Society, after which the following papers were read:

A New Method for the Separation of Arsenic, Antimony, Selenium and Tellurium from one another and from other Metals,' A. E. Knorr; Separation of Impurities in the Electrolytic Refining of Copper,' by P. de P. Ricketts; 'The Preparation of Metallic Tellurium,' Victor Lehner.

The meeting was then adjourned to take a special train to the New Jersey Zinc and Iron Company's works at Newark, N. J., where a luncheon was served, and the process of manufacture of zinc oxide was shown. Parties were also made up to visit the Wetherill Concentrator Works, Murphy Varnish Company, Lister's Agricultural Chemical Works and others.

In the evening a business session was held at the club rooms, at which reports were received from standing committees and the retiring President made his address. M. Raoul Pictet gave an interesting discourse on the Retardation of Chemical Activities at Low Temperatures.' His subject was illustrated by a lantern projection showing a piece of metallic sodium held on a steel needle and both immersed in hydrochloric acid which had been cooled to the lowest temperature obtainable by means of solidified carbon dioxide. There was no reaction between acid and sodium or the iron until a considerable rise of temperature had taken place.

The second day's session was held at Havemeyer Hall, Columbia University, at which the following papers were read:

'Measurement of Turbidity in Water,' W. P. Mason; 'The Assay of Nux Vomica,' E. R. Squibb; The Potato and Cassava Starch Industries in the United States,' H. W. Wiley; 'Notes on the Estimation of Carbohydrates,' Traphagen and Cobleigh; 'The Action of Iodine on the Fatty Amines,' J. F. Norris; On the Constitution of Some Canadian Baryto-Celestites, C. W. Volney;

was, what Sylvester was not, also a geometer. Again and again we find the pure geometric methods of Poncelet and Chasles, though, perhaps, not full assimilation of that greater one than they who has now absorbed them--von Staudt.

Cayley not only made additions to every important subject of pure mathematics, but whole new subjects, now of the most importance, owe their existence to him. It is said that he is actually now the author most frequently quoted in the living world of mathematicians. name is, perhaps, most closely linked with the word invariant, due to his great brother-in-arms, Sylvester.

His

the expectations of the most sanguine of those who had worked for it.

The attendance was not less than one hundred and fifty at any of the sessions, and among them a number of ladies, who also graced the dinner with their presence. DURAND WOODMAN. Secretary

SCIENTIFIC BOOKS. The Collected Mathematical Papers of ARTHUR CAYLEY. 4to. 13 Vols., each $6.25. Supplementary Vol., containing Titles of Papers and Index. New York, Macmillan Co. $2.50. This republication by the Cambridge University Press of Cayley's papers, in collected form, is the most fitting monument of his splendid fame.

He must ever rank as one of the greatest mathematicians of all time. Cayley exceedingly appreciated this action of the Syndics of the Press, and seven of the large quarto volumes appeared under his own editorship.

As to what these thirteen volumes contain it seems vain to attempt even a summary. They cover the whole range of pure mathematics, algebra, analysis, mathematical astronomy, dynamics, and in particular groups, quadratic forms, quantics, etc., etc.

Though abreast of Sylvester as an analyst, he

number of functions which have the property of preserving their form unaltered after any linear transformation of the variables.' His first results, relating to what we now call invariants, he published in 1845. A second set of results, relating to what Sylvester called covariants, he published in 1846. Not until four or five years later did Sylvester take up this matter, but then came such a burst of genius that after his series of publications, in 1851-4, the giant theory of Invariants and Covariants was in the world completely equipped.

The check came when Cayley, in his second Memoir on Quantics, came to the erroneous conclusion that the number of the asyzygetic invariants of binary quantics beyond the sixth order was infinite, 'thereby,' as Sylvester says, 'arresting for many years the progress of the triumphal car which he had played a principal part in setting in motion.'

The passages supposed to prove this are marked incorrect' in the Collected Mathematical Papers. But this error was not corrected until 1869 [Crelle, Vol. 69, pp. 323-354] by Gordan in his Memoir [dated 8th June, 1868] : "Beweis dass jede Covariante und Invariante einer binaeren Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist."

Cayley at once returned to the question, found

the source of his mistake, the unsuspected and so neglected interdependence of certain syzygies, and devoted his Ninth Memoir on Quantics (7th April, 1870) to the correction of his error and a further development of the theory in the light of Gordan's results.

The whole of this primal theory of invariants may now be regarded as a natural and elegant application of Lie's theory of continuous groups. The differential parameters, which in the ordinary theory of binary forms enable us to calculate new invariants from known ones, appear in a simple way as differential invariants of certain linear groups. The Lie theory may be

illustrated by a simple example. Consider the binary quadratic form

fax2+2axy +α2y2.

Applying to f the linear transformation

1

Hence a ̧α-a2, is an invariant of the form f. In the group theory it is an invariant of the group of linear homogeneous transformations (2) on the three parameters a。, ɑ1, ɑ2.

The only covariant of ƒ is known to be ƒ itself. In the Lie theory it appears as the invariant of a linear homogeneous group on five variables, x, y, a, a, a, the transformations being defined by the equations (2), together with (1) when inverted.

In general, the invariants of a binary form of degree n are defined by a linear homogeneous group on its n + 1 coefficients, its covariants by a group on n + 3 variables.

As in all problems in continuous groups, the detailed developments are greatly simplified by employing the infinitesimal transformations of the groups concerned.

It is readily proven by the group theory that all invariants and covariants are expressible in terms of a finite number of them.

This result is, however, not equivalent to the algebraic result that all rational integral invariants (including covariants) are expressible rationally and integrally in terms of a finite number of such invariants.

Twenty years ago, in my 'Bibliography of Hyper Space and Non-Euclidean Geometry' (American Journal of Mathematics, Vol. I., Nos. 2 and 3, 1878), I cited seven of Cayley's papers written before 1873:

I. Chapters in the Analytical Geometry of (n) Dimensions. Camb. Math. Jour., Vol. IV., 1845, pp. 119–127.

II. Sixth Memoir on Quantics. Phil. Trans., Vol. 149, pp. 61-90 (1859).

III. Note on Lobatchevsky's Imaginary Geometry. Phil. Mag. XXIX., pp. 231-233 (1865).

IV. On the rational transformation between two spaces. Lond. Math. Soc. Proc. III., pp.

127-180 (1869-71).

V. A Memoir on Abstract Geometry. Phil. Trans. CLX., pp. 51-63 (1870).

VI. On the superlines of a quadric surface in five dimensional space. Quar. Jour., Vol. XII., pp. 176-180 (1871-72).

VII. On

the Non-Euclidean Geometry. Clebsch Math. Ann. V., pp. 630-634 (1872). Four of these pertain to Hyper-Space, and in that Bibliography I quoted Cayley as to its geometry as follows:

"The science presents itself in two ways— as a legitimate extension of the ordinary twoand three-dimensional geometries, and as a need in these geometries and in analysis generally. In fact, whenever we are concerned with quantities connected together in any manner, and which are or are considered as variable or determinable, then the nature of the relation between the quantities is frequently rendered more intelligible by regarding them (if only two or three in number) as the coordinates of a point in a plane or in space for more than three quantities there is, from the greater complexity of the case, the greater need of such a representation; but this can only be obtained by means of the notion of a space of the proper dimensionality; and to use such a representation we require the geometry of such space.

An important instance in plane geometry has