the constant greater velocity; and, thirdly, the friction, in which heat is indeed again produced, but which, as Clausius shows*, partly passes into the sides of the vessel. In our calculation for the case in which the gas issues from a globe into an exhausted one, the first cause only requires to be taken into account; Thomson and Joule's experiments, in which all three causes cooperate, must in any case give a greater cooling, as indeed the above numbers show. We might, indeed, consider the difference of the values in the third and fifth column to be the cooling arising from the two latter causes; I think, however, that these numbers have indeed a qualitative, but only a very small quantitative value. The question here is as to the difference of two very small magnitudes; and even if in Thomson and Joule's case the absolute error in the determination of temperature is very small, it might yet relatively be considerable. The numbers of the third column are only calculated from Mariotte and Gay-Lussac's law, but these might be calculated much more accurately from Regnault's experiments. In the first place, the values are obtained by means of rough interpolation; a more accurate calculation would not be of much use, on account of the cause given in the fourth section. The formula given in § II. for carbonic acid is manifestly only approximate, since the coefficient of (h-1) is found here to be greater than at 3°, which cannot be the case if gases generally approximate to the real condition with increasing temperature. It follows from this that the differences of specific heats, in the case of carbonic acid, change regularly, yet that the differences of these series have not the desired continuity. It would have been better to calculate the formula for carbonic acid at 100° from the coefficient of expansion by means of the method of least squares. But all this only gives greater accuracy in case the fundamental formula of Mariotte's law deduced from Regnault's investigations is more accurate than the one which occurs in his memoir. It follows, further, from the theoretically deduced formulæ, that Thomson and Joule's result, that the cooling is proportional to the difference of pressure, is inaccurate. But the deviations are too small to be capable of experimental determination. It is interesting to remark that an ideal gas, one therefore to which Mayer's hypothesis applies, would undergo no cooling in the experiment with the globes, but would in Thomson and Joule's experiments; for in an ideal gas the first cause for cooling does not exist, though the two others do, as is easily intelligible. Clausius, Abhandlungen über der mechanischen Wärmetheorie, p. 107. § VIII. Conclusion. The following remarks may be made in conclusion. In defining an ideal gas, in § I. only has it been assumed that k is constant. This is enough where it is only a question of deviations from Mariotte and Gay-Lussac's law; and the deduction of the absolute zero depends on no further assumption. But where the specific heat is concerned, it must be defined more accurately whether the gas in expanding is to perform internal work or not. The assumption that in this gas no internal work is performed is at the basis of the definition of Carnot's function of temperature. Upon this depends the further development of the theory, and its application to bodies occurring in nature. An ideal gas, therefore, must satisfy the three following conditions: k is independent of pressure and temperature, and performs in expanding no internal work. Now Cohen-Stuart has shown that the second condition follows from the first and third; the assumption, therefore, that a gas satisfies the first and third but not the second, is self-contradictory. If there were an ideal gas, the question would soon be settled; but as this is not the case, the question is, whether an ideal gas as defined above can occur; or whether perhaps it is not in contradiction with other natural laws, whether known or unknown to us. If the mechanical theory of heat is to be developed in accordance with fact and not as a mere play of formulæ, it is of the greatest importance to investigate this as accurately as possible. It frequently occurs that in an analytical investigation a new magnitude is introduced to simplify calculation; this obtains its significance from its definition, and is without influence on the results of the theory. This is the case, for instance, with the potential function. Now the assumption of an ideal gas materially simplifies the development of the mechanical theory of heat; but in the present case this is neither sufficient nor is it the chief question. If the theory is to be developed in accordance with fact, it must start from premises which occur in nature, and not from such as are arbitrarily excogitated by us. Hence the ideal gas is only another name for the condition to which, according to our idea, one or more gases approximate. The occurrence of this condition has never been experimentally proved, and it is only from the known properties of gases that we can conclude with greater or less certainty that there is such a condition. We know that at a low pressure and high temperature the deviations from the ideal condition are small, but this is by no means adequate for a firm establishment of the theory. It may readily be imagined that in applying this theory to gases a small deviation is of very small importance, but of much greater in the case of solids and liquids. It is therefore of urgent necessity to investigate as accurately as possible whether such an ideal state occurs in the case of gases-not because the condition of these gases is of such importance in itself, but because the entire theory of heat, so far as it is connected with the ordinary definition of Carnot's function of temperature, stands or falls with it. The occurrence of this ideal condition is the more probable, the more the various deviations of actual gases from the ideal condition taken together converge towards nil. As regards Mariotte and Gay-Lussac's law, this follows from Regnault's investigation, and it is confirmed by the testing of the law of volumes; the values of the internal work given in § VII. show, too, that this decreases with higher temperature and lower pressure; this also follows from Thomson and Joule's experiments, which in this respect have great value. So far as previous observations permit, I believe I may conclude, from what has here been communicated, that these deviations decrease together, by which the validity of the assumption of an ideal gas acquires a confirmation. Zütphen, February 1864. XXVII. On the Black-bulb Thermometer. By Professor TYNDALL, F.R.S. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, IN N the February Number of the Philosophical Magazine, Mr. Wilson of Rugby has published an interesting letter, in which he describes an observation made by Mr. Glaisher, to the effect that the difference of reading between a black-bulb thermometer exposed to direct sunshine, and a second thermometer shaded from the sun's direct action, diminishes as we ascend in the atmosphere. This observation points to the following conclusions which have been drawn by Mr. Wilson. 1. "If the experiment could be made, the shaded and exposed thermometers outside the earth's atmosphere would show the same very low reading." 2. "All those views of the state of the surface of the moon which represent it as alternately exposed to fiery heat and intense cold will disappear. The temperature on the dark and bright halves of the half moon is the same if there is no atmosphere." "We learn," continues Mr. Wilson, one more marvellous function of our atmosphere before unsuspected. We knew that it and the aqueous vapour it bears diffused through it save us from loss of heat. We did not know that but for it, or some part of it, we should have no heat to lose. Hence, too, we learn that conclusions as to the temperatures of other planets are still more open to uncertainty, as being in this way also influenced by their atmospheres." These are very weighty inferences. They affirm that what is heat in our atmosphere, is not heat in stellar space; that something is emitted by the sun which warms a body surrounded by an atmosphere, but which is powerless to warm the body if not so surrounded. A planetary atmosphere, in short, has a power of transmuting into heat an agency which, prior to its entering the atmosphere, is not heat, and this power of transmutation, if possessed in a very high degree, might raise the most distant planet to a high temperature. I must frankly confess that I do not consider these conclusions, however fairly drawn, to be representative of natural facts. I am ignorant of the details of Mr. Glaisher's observation, and therefore not in a position to offer any explanation of it. But the following remarks, which are in substance transferred from a paper presented some time ago to the Royal Society, and ordered to be printed in the Philosophical Transactions, may have some bearing upon the question; or if not, they may bring to the notice of meteorologists a possible defect in their observations on solar radiation, which, as far as I know, has been hitherto overlooked : 1. Solar heat as it reaches us consists partly of visible and partly of invisible rays, a portion of the latter possessing very high calorific power. 2. The ordinary black-bulb thermometer absorbs the visible rays of the sun, but the black glass of the bulb may be highly transparent to the invisible radiation. 3. The invisible calorific rays, especially, augment in power as we ascend in the atmosphere; for it is those rays which suffer most diminution of intensity during their passage through the aqueous vapour of the air. 4. Hence the black-bulb thermometer must, with reference to the total radiation falling upon it, become more and more transparent as we ascend in the atmosphere. It is not an exaggerated estimate that at the limits of our atmosphere 50 per cent. of the solar heat might cross the glass of the bulb, be reflected by the mercury within it, and contribute in no degree to the heating of the thermometer. the indications of the black-bulb thermometer, as at present constructed, are delusive, and they are especially so at great elevations. I am, Gentlemen, Royal Institution, Your obedient Servant, JOHN TYNDALL. IN XXVIII. On Archdeacon Pratt's 'Figure of the Earth.' By Captain A. R. CLARKE, Royal Engineers, F.R.S., F.R.A.S.* N the application of the method of least squares to the determination of the figure of the earth, there is this anomaly, that the semiaxes are not determined by making the sum of the squares of the errors of observation or measurement a minimum. Of the existing uncertainty as to the figure of the earth, but a small portion is due to errors of measurement or observation; it is mainly due to the circumstance that the earth is irregular, and cannot be precisely represented by any ellipsoid or other simple (algebraically speaking) surface. Imagine a spheroid having its axis coincident with that of the earth's rotation, and actually intersecting the mathematical surface of the earth in various curves; if P be any point in the former surface, p that in which the normal at P meets the irregular surface, then, as far as our present knowledge extends, and for values of the axes between certain limits, Pp may be everywhere a very small quantity, and we become aware of its existence only through its variations. These appear as local displacements of the zenith, or, in other words, discrepancies in the latitudes and longitudes of astronomical stations. Now these latitude-discrepancies, or, as they are otherwise expressed, deflections or local attractions, are in their average magnitude very much larger than the probable errors of the astronomical determinations of latitude, and indeed overwhelm the errors of the geodetical operations. Moreover the local deflections become, when we attempt to ascertain the figure of the earth, inseparably mixed up with the errors of measurement and observation. Hence to determine the figure of the earth from a number of measured arcs and observed latitudes, we have to find that spheroid which, when the measured arcs are, as it were, applied to it, shall give the sum of the squares of the deflections an absolute minimum; in other words, corrections x1, x2,... have to be applied to the observed latitudes of the points P1, P2,... such as to bring them into accordance with the spheroidal surface, while the sum of the squares of the quantities x1, x2,... is to be an absolute minimum. The spheroid * Communicated by the Author. |