depth of inspiration. A decrease in quantity was caused by sp. ammon. co. (3iss), sp. ammon. fæet. (3iss), tincture of opium (20 m), morphia (and gr.), tartarized antimony (gr.), and chloride of sodium.

Carbonate of ammonia (15 grains) caused a small increase at first and then a small decrease; febrifuge medicines had a like effect. Chloroform (25 m and 3ss), by the stomach, varied the quantity from an average increase of 28 cub. ins. to an average decrease of 20 cub. ins. per minute; with a maximum increase of 63 cub. ins. per minute. Chloric ether (3ss) also varied the quantity, but there was an average increase of 17 cub. ins. per minute, and of 1.8 per minute, in the rate; whilst the pulse fell on the average 1·7 per min. Chloroform, by inhalation (to just short of unconsciousness), lowered the quantity a little during the inhalation, and more so afterwards. The rate was unchanged, but the pulse fell, on an average, 1.7 per min. Amylene similarly administered and to the same degree, increased the quantity during inhalation 60 cub. ins. per min., but afterwards decreased it to 100 cub. ins. per min. less than during the inhalation. The rate of respiration was unchanged: the pulse fell 6 per min. at the end of the observation.

Digitalis (infusion 3i) varied the quantity, increasing it at first and then decreasing it. The rate of inspiration was unaffected, whilst that of pulsation somewhat increased.

The paper is accompanied by tables of numerical statements, and by diagrams exhibiting the results in a series of curves.

May 7.-The Lord Wrottesley, President, in the Chair.

The following communications were read:

"On the Plasticity of Ice, as manifested in Glaciers." By James Thomson, A.M., C.E. Belfast.

The object of this communication is to lay before the Royal Society a theory which I have to propose for explaining the plasticity of ice at the freezing-point, which is shown by observations by Professor James Forbes, and which is the principle of his Theory of Glaciers.

This speculation occurred to me mainly in or about the year 1848. I was led to it from a previous theoretical deduction at which I had arrived, namely, that the freezing-point of water, or the meltingpoint of ice, must vary with the pressure to which the water or the ice is subjected, the temperature of freezing or melting being lowered as the pressure is increased. My theory on that subject is to be found in a paper by me, entitled "Theoretical Considerations on the Effect of Pressure in Lowering the Freezing-Point of Water," published in the Transactions of the Royal Society of Edinburgh, vol. xiv. part 5, 1849*. It is there inferred that the lowering of the freezingpoint, for one additional atmosphere of pressure, must be 0075° Centigrade; and that if the pressure above one atmosphere be denoted in atmospheres as units by n, the lowering of the freezing-point,

*The paper here referred to is also to be found in the Cambridge and Dublin Mathematical Journal for November 1850 (vol. v. p. 248), where it was republished with some slight alterations made by myself.

denoted in degrees Centigrade by t, will be expressed by the formula t='0075 n.

The phenomena which I there predicted, in anticipation of direct observations, were afterwards fully established by experiments made by my brother, Professor William Thomson, and described in a paper by him, published in the Proceedings of the Royal Society of Edinburgh (Feb. 1850) under the title, "The Effect of Pressure in lowering the Freezing-Point of Water experimentally demonstrated*."

The principle of the lowering of the freezing-point by pressure being laid down as a basis, I now proceed to offer my explanation, derived from it, of the plasticity of ice at the freezing-point as follows:

If to a mass of ice at 0° Centigrade, which may be supposed at the outset to be slightly porous, and to contain small quantities of liquid water diffused through its substance, forces tending to change its form be applied, whatever portions of it may thereby be subjected to compression will instantly have their melting-point lowered so as to be below their existing temperature of 0° Cent. Melting of those portions will therefore set in throughout their substance, and this will be accompanied by a fall of temperature in them on account of the cold evolved in the liquefaction. The liquefied portions being subjected to squeezing of the compressed mass in which they originate, will spread themselves out through the pores of the general mass, by dispersion from the regions of greatest to those of least fluid pressure. Thus the fluid pressure is relieved in those portions in which the compression and liquefaction of the ice had set in, accompanied by the lowering of temperature. On the removal of this cause of liquidity-the fluid pressure, namely,--the cold which had been evolved in the compressed parts of the ice and water, freezes the water again in new positions, and thus a change of form, or plastic yielding of the mass of ice to the applied pressures, has occurred. The newly-formed ice is at first free from the stress of the applied forces, but the yielding of one part always leaves some other part exposed to the pressure, and that, in its turn, acts in like manner; and, on the whole, a continual succession goes on of pressures being applied to particular parts-liquefaction in those partsdispersion of the water so produced, in such directions as will relieve its pressure, and recongelation, by the cold previously evolved, of the water on its being relieved from this pressure. Thus the parts recongealed after having been melted must, in their turn, through the yielding of other parts, receive pressures from the applied forces, thereby to be again liquefied, and to enter again on a similar cycle of operations. The succession of these processes must continue as long as the external forces tending to change of form remain applied to the mass of porous ice permeated by minute quantities of water.

Postscript received 22nd April, 1857.

It will be observed that in the course of the foregoing communica

*The paper by Prof. William Thomson, here referred to, is also to be found republished in the Philosophical Magazine for August 1850.

tion, I have supposed the ice under consideration to be porous, and to contain small quantities of liquid water diffused through its substance. Porosity and permeation by liquid water are generally understood, from the results of observations, and from numerous other reasons, to be normal conditions of glacier ice. It is not, however, necessary for the purposes of my explanation of the plasticity of ice at the freezing-point, that the ice should be at the outset in this condition; for, even if we commence with the consideration of a mass of ice perfectly free from porosity, and free from particles of liquid water diffused through its substance, and if we suppose it to be kept in an atmosphere at or above 0° Centigrade, then, as soon as pressure is applied to it, pores occupied by liquid water must instantly be formed in the compressed parts in accordance with the fundamental principle of the explanation which I have propounded -the lowering, namely, of the freezing- or melting-point by pressure, and the fact that ice cannot exist at 0° Cent. under a pressure exceeding that of the atmosphere. I would also wish to make it distinctly understood that no part of the ice, even if supposed at the outset to be solid or free from porosity, can resist being permeated by the water squeezed against it from such parts as may be directly subjected to the pressure, because the very fact of that water being forced against any portions of the ice supposed to be solid will instantly subject them to pressure, and so will cause melting to set in throughout their substance, thereby reducing them immediately to the porous condition.

Thus it is a matter of indifference as to whether we commence with the supposition of a mass of porous or of solid ice.

"On the Comparison of Transcendents, with certain applications to the Theory of Definite Integrals." By George Boole, Esq., Professor of Mathematics in Queen's College, Cork.

The following objects are contemplated in this paper:

1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation.

2ndly. The application of that theorem to the comparison of algebraical transcendents.

3rdly. Its application to the comparison of functional transcendents, i. e. of transcendents in the differential expression of which an arbitrary functional sign is involved.

4thly. Certain extensions of the theory of definite integrals both single and multiple, founded upon the results of the application last mentioned.

In the expression of the fundamental theorem for the summation of integrals, the author introduces a symbol, ✪, similar in its definition to the symbol employed by Cauchy in the Calculus of Residues, but involving an additional element. The interpretation of this symbol is not arbitrary, but is suggested by the results of the investigation by which the theorem of summation is obtained. All the general theorems demonstrated in the memoir either involve this symbol in their expression, or are immediate consequences of theorems into the expression of which it enters.

The author directly applies his theorem of summation both to the solution of particular problems in the comparison of the algebraical transcendents, and to the deduction of general theorems. Of the latter the most interesting, but not the most general, is a finite expression for the value of the sum

where $ and denote any rational functions of x; the equation by which the limits of the integrals are determined being of the form

773

"=x, in which Χ is also a rational function of x.

The forms of 9, 4, and x are quite unrestricted, except by the condition of rationality. Previous known theorems of the same class, such as Abel's, suppose a polynomial and specify the form of p. In the author's result, the rational functions p, 4, and x are not decomposed. In a subsequent part of the paper, after investigating a general theorem applicable to the summation of all transcendents which are irrational from containing under the sign of integration any function which can be expressed as a root of an equation whose coefficients are rational functions of x, he explains by means of it, the cause of the peculiarity above noticed.

In the section on functional transcendents, a remarkable case presents itself in which the several integrals under the sign of summation, Σ, close up, if the expression may be allowed, into a single integral taken between the limits of negative and positive infinity. The result is an exceedingly general theorem of definite integration, by means of which it is demonstrated, that the evaluation of any definite integral of the form

in which (x) is a rational function of x, and in which a, a,.. a, are positive, and A,, Ag. A, are real, the number of those constants being immaterial, may be reduced to the evaluation of a definite integral of the form

in which (v) is a rational function of v of the same order of complexity as the function (x). Two limited cases of this theorem are referred to as already known,-one due to Cauchy, the other published by the author some years ago.

The remainder of the paper is occupied with applications of the above general theorem of definite integration. Of the Notes by which the paper is accompanied, the first discusses the connexion between the author's symbol and Cauchy's, and contains two theorems, one exhibiting the general solution of linear differential equations with constant coefficients, the other the general integral of rational fractions. Both these theorems involve in their expression the symbol . The second Note is devoted to the interpretation of some theorems for the evaluation of multiple integrals, investigated in the closing section of the paper.

LXIV. Intelligence and Miscellaneous Articles.

ON A NEW POLARIZER OF ICELAND SPAR.—EXPERIMENT ON

FLUORESCENCE. BY LÉON FOUCault.

WHEN the object is to polarize a pencil of white light completely,

the best known method is to have recourse to the use of the Nicol's prism; but for operating upon a sheaf of a certain volume, of 4 to 5 centimetres in diameter, for example, the Nicol's prism becomes expensive and difficult to procure, in consequence of the rarity of fine specimens of spar.

The cutting adopted in the construction of the Nicol's prism is necessarily attended with a great expense of material. The prism to be complete, must be taken from a piece of spar the longitudinal edges of which are at least equal to three times one of the equal sides which terminate the bases. The piece is then cut from one obtuse angle to the other, through a plane inclined at 88° upon the plane of the bases, and perpendicular to the plane of their small diagonals. The two faces thus obtained are polished, and fastened together by means of Canada balsam.

When a parallelopiped thus formed is turned towards a uniformly illuminated ground, and we look through the piece in the direction of its axis, a field of polarization is seen included between two curved bands, one red, and the other blue, which correspond with the limit directions in which the ordinary and extraordinary rays are transmitted. These bands enclose an angular space of 32°, which renders the Nicol's prism an analyser applicable in all circumstances when the inclination of the rays, which are to be observed simultaneously, does not exceed 32°.

But this angular extent of the field of polarization, which is prized in the Nicol's prism considered as an analyser, no longer possesses the same interest when the apparatus is simply to play the part of a polarizer; for in that case the action which is to be produced in general only affects a pencil of light of nearly parallel rays. So that, under such circumstances, there would be an advantage in increasing the transverse dimensions of the prism, even when this would cause a certain reduction in the extent of the angular field of polarization. In reflecting on the data of the question, I have in fact ascertained that the cutting of the Nicol's prism may be modified so as to diminish its length considerably without injury to the effects which it may produce as a polarizer.

I take a parallelopiped of spar, of which the longitudinal edges are only equal to five-fourths of one of the sides of the bases; I carry from one obtuse angle to the other a section inclined at 59° upon the plane of the bases, and the new faces being polished, I replace the two fragments in their natural position without glueing them, and taking care to preserve between the new faces a little space in which air remains, and which, under suitable incidence, causes the complete reflexion of the ordinary ray.

On looking through a rhomb cut in this way, and mounted otherwise like a Nicol's prism, we again find that there exists an angular field of polarization; but the index of refraction of the air being considerably less than those of the two rays propagated by the spar,