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there is no overlapping of these plates in the other genera referred to this group, it includes two types of structure. The author then discussed the characters presented by the test in the genera of the Perischoechinidae (namely Archæocidaris, Palachinus, Perischodomus, Lepidechinus, Eocidaris, Melonites, and Oligoporus), and pointed out that although we have no conclusive evidence of the presence of membranous interspaces along with the overlapping plates in Archæocidaris, the fragmentary condition in which the remains of that form are usually found would lead us to infer their existence. No known paleozoic genus exhibits the want of distinction between the ambulacra and interambulacra on the ventral half of the test seen in the recent genus Phormosoma. In Melonites and Oligoporus the author described an increase in the number of rows of plates in the ambulacra, and he indicated that all the Perischoechinidæ differ from the later Echini by the increased number of perforations in the ocular and genital plates.
2. "On the discovery of Foraminifera &c. in the Boulder-clays of Cheshire." By William Shone, Esq., jun.
In this paper the author described the occurrence of Foraminifera, Entomostraca, and some other small organic bodies in the Boulderclay at Newton by Chester and at Dawpool. They were found partly in the interior of specimens of Turritella terebra, and partly free in the Boulder-clay; but those obtained from the Turritella were in better condition than the others. The Foraminifera generally agree precisely with those found in the tidal parts of the river Dee. The author stated further that the Turritella containing Foraminifera are filled with a fine greyish-white sand, in which the minute fossils abound; and he discussed the probable conditions under which the deposit containing them had been formed.
3. “On the occurrence of a Tremadoc area near the Wrekin in South Shropshire, with description of a new Fauna." By Charles Callaway, Esq., M.A., B.Sc., &c.
The author stated that in an exposure of light-green micaceous shales dipping south-east at 50° at Shineton near Cressage, which are represented as of Caradoc age in the Geological-Survey Map, he found a series of Trilobites and other fossils which induced him to regard these Shineton shales as belonging to the Lower Tremadoc series. He described as new species Asaphus Eos, Conocoryphe Salteri, C. angulifrons, Platypeltis Croftii, Conophrys salopiensis, Lichapyge cuspidata, Lingulella Nicholsoni, Metoptoma Sabrina, and Theca lineata. The author regarded these shales as the equivalents of beds containing Dictyonema found near Malvern and at Pedwardine.
XX. Intelligence and Miscellaneous Articles.
ON THE EXPRESSION OF THE WORK RELATIVE TO AN ELEMENTARY TRANSFORMATION. BY J. MOUTIER.
CLAUSIUS has recently given a demonstration of Carnot's theorem, founded on the expression of the work relative to an elementary transformation on the hypothesis now generally admitted, in which heat is considered to be a mode of motion. M. Ledieu arrived at the same result by a different path. These solutions leave the nature of the motion undetermined, and by that very fact present the greatest generality. My intention is to treat the same question by admitting that heat consists of a vibratory motion. The analogy existing between heat and light permits us to suppose that it is so; and as the vibration theory suffices for the explanation of all optical phenomena, there is reason to investigate whether it can likewise account for the phenomena of heat. This particular hypothesis is not new in science. It is true that it restricts the generality of the solution; but, on the other hand, it permits us to state precisely the nature of certain phenomena.
The vibratory motion with which each point is animated can be decomposed according to three rectangular directions; each component motion is an oscillatory rectilinear motion of the same period, produced by a force proportional to the distance from the material point to a fixed centre.
If we represent by m the mass of the material point, by the acceleration at the unit of distance, by a the amplitude of the oscillation, the mean value of the force is f=mpa. 27
The duration i of an oscillation is i=
The maximum velocity of the material point is U= 2
The maximum semi-vis viva is mU2=fa.
The mean semi-vis viva mu2 is the half of the maximum semivis viva; mu2=fa. This mean is considered to be proportional to the absolute temperature T.
The elementary work which corresponds to a rise of temperature dT is composed of two parts: one is equal to half the increment of the mean vis viva; the other arises from modifications brought into the vibratory motion by supposing that the mean vis viva preserves the same value, or else that the temperature remains constant.
The temperature remaining constant, the amplitude of the oscillation may change, provided that the duration of an oscillation varies in the same ratio. If the amplitude of the oscillation is increased by the quantity da, there results a work which is expressed by the product of the mean value of the force into the increment of the amplitude, or fda.
Now, as the ratio
is to remain constant,
The portion of the work relative to an elementary transformation is therefore, for the motion we are considering,
The same reasoning applies to each of the three rectangular directions in which the motion of the material point is projected. The work dL, relative to an elementary transformation, is the sum of the quantities analogous to the preceding; so that, calling mv2 the mean vis viva of a material point, we shall have for the entire system,
We thus find again the expression given by Clausius. If we represent by M the weight of the body, by k its absolute specific heat, by E the mechanical equivalent of the heat, on the hypothesis adopted concerning heat,
The value of the elementary work dL can then be put under the form
Admitting, as before, that heat consists in a vibratory motion, various phenomena can be analyzed from that point of view.
1. With bodies in the solid state the ordinary specific heat is sensibly equal to three times the absolute specific heat. Let us see what indication is furnished in regard to this by the above-stated theory.
The quantity of heat necessary to increase the temperature of the body by dT is then 3MKɗT. A part of this heat, MKdT represents the increment of the heat really existing within the body; so that the heat expended in work is 2MKdT. We have therefore
Substituting for i and T their values deduced from the preceding relations, we find the condition f= constant.
Therefore, in bodies in the solid state, when the vulgar specific heat is equal to triple the absolute specific heat, the molecular forces have a sensibly constant value, independent of the temperature. We thus rediscover a property enunciated in a previous memoir*.
* Comptes Rendus, vol. lxxi. p. 934; Annales de Chimie et de Physique, S. 4. vol. xxiv. p. 306.
2. Let us seek as well the condition for there being no heat consumed in internal work when the body is heated under constant volume, which is sensibly the case for permanent gases.
The quantity of heat necessary for raising the temperature by T is then KdT; in this case dL=0,
This relation is equivalent to the following
By substituting for i and T the values deduced from the preceding relations we find for the condition
Therefore, that there may be no heat consumed in internal work when a body is heated under constant volume, the amplitude of the oscillations must remain the same; on the contrary, there is expenditure of heat in internal work when the amplitude of the oscillations increases.
3. When the body undergoes transformation with the temperature constant, as in changes of state, the quantity of heat necessary for effecting the transformation is
designating by i, and i the durations of an oscillation before and after the transformation, and by log the Napierian logarithm. The temperature being the same, if a, and a denote the corresponding durations of an oscillation,
But, besides, f, and ƒ denoting the mean values of the molecular forces before and after the transformation, the temperature remaining the same,
This relation shows the connexion existing between the heat necessary for determining a change of state, such as fusion or vaporization, and the variation of the molecular actions in consequence of the change of state. When the molecular actions diminish (which is the ordinary case), the transformation demands an expenditure of heat (heat of fusion or vaporization); while the body evolves heat when the change of state is accompanied by augmentation of the molecular forces.-Comptes Rendus de l'Académie des Sciences, vol. lxxx. pp. 40–44.
ON THE ANALOGIES PRESENTED BY THE LIBERATION OF GASES FROM THEIR SUPERSATURATED SOLUTIONS AND THE DECOMPOSITION OF CERTAIN EXPLOSIVE BODIES. BY D. GERNEZ.
I established, long since, that, in supersaturated gaseous solutions, the excess of the quantity of gas dissolved above the normal quantity (that is, above that which the liquid would dissolve in the same conditions of temperature and pressure) does not escape, if no mechanical action be interposed, unless any gaseous atmosphere (retained, for example, at the surface of a solid body or in the capillary cavities of a porous substance) be introduced into the interior of the liquid. It is in this atmosphere, which plays the part of a vacuum in relation to the different gas dissolved, that the latter escapes at the free surface of the liquid. Now the sides of vessels often retain, even when they appear wet, a gaseous layer localized especially in the anfractuosities which are almost always found at the surface of solid bodies. Hence it results that, in vessels which have not undergone special preparation, supersaturated gaseous solutions produce in more or less abundance bubbles of gas upon the sides. But if, by washing successively with potass, boiling distilled water, and alcohol, the superficial layer of glass vessels (in certain points of which a small quantity of air is held) be carefully dissolved, we find that not a single gas-bubble will form on the side bathed by the liquid, no more than in the interior of the supersaturated solution, between very wide limits of temperature and pressure.
The emission of the gas then takes place only at the free surface of the liquid; exchanges are made from layer to layer with a slowness such that, for instance, water saturated with carbonic acid under a pressure of about 2 atmospheres, and exposed in an open tube at temperatures near 8° C., is still supersaturated in the layer situated at 10 centims. from the surface, even after fifty days.
When the pressure is lessened the gas is still emitted only at the surface, if the vessel has been properly prepared. Thus water saturated with carbonic acid under a higher pressure than 2 atmospheres has been very easily kept in the vacuum made with the mercury pump, without one bubble of gas being disengaged at the interior of the solution; and yet the manometer of the receiver indicated only a pressure equal to the maximum tension of aqueous vapour at the temperature of the experiment. The gas escaped only at the surface, without a bubble appearing, and with a relatively feeble velocity.
If a gaseous atmosphere be introduced into this solution at the surface of which a vacuum is maintained, a lively effervescence is produced which resembles violent ebullition. I have realized the experiment by plunging into Seltzer water a fragment of platinumsponge or of binoxide of manganese held at the end of a platinum wire all the liquid above the porous substance was violently projected, while below not a bubble of gas was liberated.
* Comptes Rendus, Nov. 19, 1866, vol. Ixiii. p. 883.