So much must suffice to give an idea of the object and scope of this small volume, which, it should be said, presupposes "a familiarity with the elements of algebra and trigonometry." To students thus qualified "its subject-matter and treatment constitute a rapid review of the underlying principles of those subjects, including in its most general aspects the algebra of complex quantities."

Elasticity. By Dr. B. WILLIAMSON, F.R.S.

(London: Longmans, 1894; pp. x+135.)

J. J. W.

THE full title, viz. 'Introduction to the Mathematical Theory of the Stress and Strain of Elastic Solids,' indicates that this little book is an elementary one, and that it does not treat of the more advanced portions of the general Mathematical theory. These, as our readers know, are admirably discussed in Mr. Love's fuller treatise. The subject is of the highest practical importance, and we are glad to have such an Introduction drawn up by so competent a writer as Dr. Williamson. Our author states at the outset that he adopts the notation suggested years ago by the late Professor Townsend: a notation which "has the advantage of harmonizing with the generally recognized method of representing the equation of a surface of the second degree." The discussion is in the main confined to the consideration of perfectly elastic solids, so that the results are, of course, only approximately true for actual substances.

The work is divided into five chapters. In Chapter I., Strain is treated of under the heads of Homogeneous and Heterogeneous Strain, and Strain in Curvilinear Coordinates. Chapter II. is devoted to Homogeneous and Heterogeneous Stress. The connexion between Stress and Strain is the subject of Chapter III., and occupies three sections, on Work and Potential Energy, Case of Isotropic Substances, and Applications. The Torsion of Prisms is considered in Chapter IV., and Elastic Beams in Chapter V. From this summary it will be seen that a fair amount of interesting ground is covered. That the text is clearly put goes without saying, but there is no great scope for novelty of treatment. Dr. Williamson takes care to give references, where called for, to recent memoirs and treatises bearing on the subject. It only remains to add that each chapter is closed with a selection of good illustrative exercises from College Problem papers, and that there is an index at the close of the volume.

On the Definitions of the Trigonometric Functions. By A. MACFARLANE, D.Sc. (Boston: J. S. Cushing & Co.; 49 pp.) THIS is a paper the substance of which was communicated to the Mathematical Congress at Chicago on the 22nd August in last year. It is a following up of previous papers by the same author on points connected with Space Analysis. Three of these are respectively entitled Principles of the Algebra of Physics,' in which Dr. Macfarlane introduced a certain trigonometric notation

for the partial products of two vectors. This notation has been discussed by Mr. Heaviside in the Electrician' (December 9th, 1892), by Prof. Alfred Lodge in Nature' (November 3rd, 1892) and elsewhere. The present paper is in part a rejoinder to the statements of these gentlemen. Then followed a paper on The Imaginary of Algebra,' and 'The Fundamental Theorems of Analysis Generalized for Space.'

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The author here proposes to review critically the historical definitions of the trigonometric terms, and the definitions, triangular, circular, or hyperbolic, given in the best modern treatises at my command; then to devise a logical system of definitions which will apply to space-analysis and that modern trigonometry, which, as Prof. Greenhill shows (Diff. and Int. Calc. p. 61), includes the properties both of circular and hyperbolic functions, and will be able to bring within the same domain the properties of the elliptic, general hyperbolic, and other functions."

This wide extent is not gone over here, but attention is mainly given to Plane Trigonometry: Trigonometry in space is handled in a further pamphlet entitled "The Principles of Elliptic and Hyperbolic Analysis.'

Dr. Macfarlane discusses in detail the treatment adopted by De Morgan and more recently by Dr. Hobson, Messrs. Levett and Davison, and Mr. Hayward (in his Vector Algebra and Trigonometry'). M. Laisant's Essai sur les fonctions hyperboliques is noted with warm approval, and then Dr. Macfarlane proceeds to show how the "several species of trigonometric functions-the triangular, the circular, and the ex-circular " (using Mr. Hayward's nomenclature) "may be defined in harmony with one another. The method adopted is afterwards shown to be applicable to the logarithmic spiral, ellipse, and general hyperbola, and to a mixed curve composed partly of a circle, partly of an ex-circle." In the pamphlet which follows the one before us the method is said to be applied to ellipsoidal and hyperboloidal trigonometry.

To the Editors of the Philosophical Magazine.

GENTLEMEN,

Will you kindly allow me to offer a word of explanation respecting the notice of Wiedemann and Ebert's Practical Physics,' which appeared in the March number of the Philosophical Magazine (p. 334). In writing the notice I had occasion to refer to the treatise by Prof. F. Kohlrausch as seeming "somewhat antiquated in a modern laboratory." It should be stated, in justice to Prof. Kohlrausch, that this remark was intended to be applied only to the earlier editions of his work; in a recently published German edition the text has been revised and many new experiments have been added. A similar revision of the English translation is greatly to be desired.

University College, Liverpool,
April 16th.

I remain, Gentlemen,
Yours truly,

JAMES L. HOWARD.

LI. Intelligence and Miscellaneous Articles.

ON THE THERMAL BEHAVIOUR OF LIQUIDS. To the Editors of the Philosophical Magazine. GENTLEMEN,

IN

N answer to the letters of Drs. B. Galitzine and P. de Heen in the April number of this Journal, we beg to make the following remarks:

:

It is quite true, as stated by Dr. Galitzine, that our criticism of the arrangements employed to obtain constant high temperatures does not apply to him (provided that the naphthaline employed by him as a jacketing vapour was pure). This, however, we pointed out in our paper, at the same time calling attention to the fact that the method was, with certain modifications, that employed and recommended for many years by ourselves. We think, however, that Dr. Galitzine's results are open to criticism on the following grounds :--(1) The complete elimination of alcohol from ether is not an easy matter, and cannot be effected by the action of metallic sodium; but no other method of purification is mentioned in his paper. The last trace of alcohol may be removed by shaking the ether twenty or thirty times with small quantities of water, or by repeated distillation over phosphorus pentoxide; but we do not know of any other method. (2) Although we fully appreciate the great pains taken by Dr. Galitzine to purify his ether from air, yet we doubt whether any method involving the sealing of a glass tube is capable of giving perfectly satisfactory results. In the Trans. Chem. Soc. for 1891, p. 37, a new method of determining the specific volumes of liquids and of their saturated vapours was described by one of us; and experience has shown that the chief difficulty in this method is the necessity of sealing the tube containing the liquid. The vapour of the liquid may undergo slight decomposition owing to the high temperature, or a trace of air may be expelled from the glass. It has been frequently observed in the experiments carried out by this method that when a minute quantity of air or permanent gas is present, the attainment of the final state of equilibrium is greatly retarded. It has also been observed that the retardation increases with the amount of permanent gas present, and that if there is a comparatively large amount (though actually a very small one), the results are rendered entirely untrustworthy.

The influence of the permanent gas has also been found to increase rapidly as the critical point is approached, though it may be very marked at much lower temperatures. This question has been shortly discussed by one of us (Trans. Chem. Soc. 1891, p. 128), and very fully by Dr. J. P. Kuenen (Verslagen der Afdeeling Naturkunde der Kon. Akademie, Leiden, April 1892, p. 422; June 1892, p. 15; Oct. 1893, p. 85); and we fully agree with Dr. Kuenen in the great importance he attaches to retardation.

Dr. de Heen requests us to point out the errors of reasoning or experimentation that he has made in the demonstration of his fundamental proposition, and asks for an explanation why we characterize his paper as "very inaccurate."

Our criticism, which refers to the experimental work on which the reasoning is based, was caused by the very wide discrepancies between the critical temperatures observed by Dr. de Heen and by other experimenters. As examples we may cite the following:

With such differences in the most easily determined critical constant it seemed to us to be useless to discuss the much more difficult observations on volumes.

We are still of opinion that the interpretation of M. Gouy may be admitted, and we think, with Dr. Kuenen, that the reason why a stable equipoise in a vertical tube is not immediately produced is due to the presence of impurity, in most cases of air or other permanent gas.

W. RAMSAY.
S. YOUNG.

SUPPLEMENTARY REMARKS ON CHANGES OF TEMPERATURE

CAUSED BY CONTACT OF LIQUIDS WITH SILICA†. BY DR. G. GORE, F.R.S.

In addition to the remarks already made on the results of the experiments in the above research, I beg leave to offer a further explanation of the phenomena, and to say that all such changes of temperature caused by the mere contact of liquids with solids are, in my opinion, consistent with the statement that whenever two substances approach or touch each other, they lose energy; and when they mutually recede, the opposite effect occurs.

That this statement appears to be true of all bodies, whether they are similar or dissimilar, of atoms, molecules, and masses, is shown in many ways, only a few of which need be mentioned. It is shown as loss of potential mechanical energy by falling bodies, * The letters A and B refer to two different specimens. + See Philosophical Magazine, March 1894, pp. 306–316.

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usually as loss of heat when substances of any kind are compressed, when liquids "adhere" to solids (as in the above experiments), and when substances chemically combine;" it is also shown, though in a less simple form, when bodies approach each other by electric and magnetic "attraction." Conversely, when a gas is rarefied by mechanical means, it gains heat by absorption; similarly, when a film of air adhering to a solid is displaced by a liquid, and when the molecules of a solid or a liquid are separated farther asunder by a solution in a liquid without chemical union, or by dilution of its solution, heat is usually absorbed.

Although in every one of the 98 experiments of the above research heat was evolved and lost, in about half of the 58 trials with silica the dissolved substance appeared to absorb heat by contact with the powder. These apparent exceptions were probably due to the amount of heat absorbed by the receding film of air being greater than that evolved by the approach and contact of the dissolved substances, and thus those cases were made to appear inconsistent with the above general statement. Similarly, instances of electric and magnetic "repulsion" are probably only apparent exceptions to that statement, the real phenomena being obscured by the attendant circumstances.

As in all these various cases, whether of "adhesion" of films of air, water, or dissolved substance to solids, of chemical union, or of electric and magnetic attraction, the only factors of the mechanical energy of the molecules are mass and velocity, and as in every instance the mass of neither of the acting substances is perceptibly altered, the loss of mechanical energy of bodies by mutual approach and contact, and the gain by mutual recession, must consist entirely of velocity. It is well known that a solid when dissolved in a liquid behaves like a gas, and as the molecules of a gas move faster the better the exhaustion" (Crookes, Phil. Trans. Roy. Soc. 1879, p. 160), so do those of a liquid by being separated by dilution; this increase of molecular velocity by dilution is also shown by an increase of electromotive force (see Proc. Birm. Phil. Soc. vol. viii. pp. 63–138).

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As these supplementary remarks appear to afford a much wider and more complete theoretical explanation of the phenomena observed with silica &c. than that offered in the original communication, I venture to submit them for consideration.

POLARIZATION OF NON-DIFFRACTED INFRA-RED RADIATION BY WIRE GRATINGS. BY H. DU BOIS AND H. RUBENS.

The results of this investigation, a preliminary account of which was given at the Edinburgh B. A. Meeting (1892), may be summarized as follows:

Gratings were finally made of metal wire down to a diameter of 2.5 mcm. (" millicentimetre "=0·001 cm.), the interspaces between Phil. Mag. S. 5. Vol. 37. No. 228. May 1894.

2 M