the relations of the specific heats, because the knowledge of the specific heats of gases at that time was of so uncertain a character. He attributed most weight to his later determinations of the mechanical equivalent made by the direct method of friction of liquids. He showed that the results obtained with different liquids, water, mercury and sperm oil, were the same, namely, 782 foot-pounds; and finally repeating the method with water, using all the precautions and improvements which his experience had suggested, he obtained the value 772 foot-pounds, which was accepted universally for many years, and has only recently required alteration on account of the more exact definition of the heat unit, and the standard scale of temperature (see CALORIMETRY). The great value of Joule's work for the general establishment of the principle of the conservation of energy lay in the variety and completeness of the experimental evidence he adduced. It was not sufficient to find the relation between heat and mechanical work or other forms of energy in one particular case. It was necessary to show that the same relation held in all cases which could be examined experimentally, and that the ratio of equivalence of the different forms of energy, measured in different ways, was independent of the manner in which the conversion was effected and of the material or working substance employed. As the result of Joule's experiments, we are justified in concluding that heat is a form of energy, and that all its transformations are subject to the general principle of the conservation of energy. As applied to heat, the principle is called the first law of thermo-dynamics, and may be stated as follows: When heat is transformed into any other kind of energy, or vice versa, the total quantity of energy remains invariable; that is to say, the quantity of heat which disappears is equivalent to the quantity of the other kind of energy produced and vice versa. The number of units of mechanical work equivalent to one unit of heat is generally called the mechanical equivalent of heat, or Joule's equivalent, and is denoted by the letter J. Its numerical value depends on the units employed for heat and mechanical energy respectively. The values of the equivalent in terms of the units most commonly employed at the present time are as follows: 777 foot-pounds (Lat. 45°) are equivalent to 1 B. Th. U. (b deg.Fahr.) 426-3 kilogrammetres 1 kilogram-deg.C. or kilocalorie. 426-3 grammetres 4.180 joules " I gram-deg. C. or calorie. 1 gram-deg. C. or calorie. The water for the heat units is supposed to be taken at 20° C. or 68° F., and the degree of temperature is supposed to be measured by the hydrogen thermometer. The acceleration of gravity in latitude 45° is taken as 980-7 C.G.S. For details of more recent and accurate methods of determination, the reader should refer to the article CALORIMETRY, where tables of the variation of the specific heat of water with temperature are also given. The second law of thermodynamics is a title often used to denote Carnot's principle or some equivalent mathematical expression. In some cases this title is not conferred on Carnot's principle itself, but on some axiom from which the principle may be indirectly deduced. These axioms, however, cannot as a rule be directly applied, so that it would appear preferable to take Carnot's principle itself as the second law. It may be observed that, as a matter of history, Carnot's principle was established and generally admitted before the principle of the conservation of energy as applied to heat, and that from this point of view the titles, first and second laws, are not particularly appropriate. though probably correct, contain an undetermined function (Carnot's F't, Clapeyron's 1/C) of the temperature. He determines the value of this function to be J/T by assuming, with Séguin and Mayer, that the work done in the isothermal expansion of a gas is a measure of the heat absorbed. From the then accepted value 078 of the difference of the specific heats of air, he finds the numerical value of J to be 374 kilogrammetres per kilo-calorie. Assuming the heat equivalent of the work to remain in the gas, he obtains expressions similar to Clapeyron's for the total heat and the specific heats. In consequence of this assumption, the formulae he obtained for adiabatic expansion were necessarily wrong, but no data existed at that time for testing them. In applying his formulae to vapours, he obtained an expression for the saturation-pressure of steam, which agreed with the empirical formula of Roche, and satisfied other experimental data on the supposition that the co-efficient of expansion of steam was 00423, and its specific heat 1.69-values which are now known to be impossible, but which appeared at the time to give a very satisfactory explanation of the phenomena. The essay of Hermann Helmholtz, On the Conservation of Force (Berlin, 1847), discusses all the known cases of the transformation of energy, and is justly regarded as one of the chief landmarks in the establishment of the energy-principle. Helmholtz gives an admirable statement of the fundamental principle as applied to heat, but makes no attempt to formulate the correct equations of thermodynamics on the mechanical theory. He points out the fallacy of Holtzmann's (and Mayer's) calculation of the equivalent, but admits that it is supported by Joule's experiments, though he does not seem to appreciate the true value of Joule's work. He considers that Holtzmann's formulae are well supported by experiment, and are much preferable to Clapeyron's, because the value of the undetermined function F't is found. But he fails to notice that Holtzmann's equations are fundamentally inconsistent with the conservation of energy, because the heat equivalent of the external work done is supposed to remain in the gas. That a quantity of heat equivalent to the work performed actually disappears when a gas does work in expansion, was first shown by Joule in the paper on condensation and rarefaction At the conclusion of this paper of air (1845) already referred to. he felt justified by direct experimental evidence in reasserting definitely the hypothesis of Séguin (loc. cit. p. 383) that "the steam while expanding in the cylinder loses heat in quantity exactly proportional to the mechanical force developed, and that on the condensation of the steam the heat thus converted into power is not given back." He did not see his way to reconcile this conclusion with Clapeyron's description of Carnot's cycle. At a later date, in a letter to Professor W. Thomson (Lord Kelvin) (1848), he pointed out that, since, according to his own experiments, the work done in the expansion of a gas at constant temperature is equivalent to the heat absorbed, by equating Carnot's expressions (given in § 17) for the work done and the heat absorbed, the value of Carnot's function F't must be equal to J/T, in order to reconcile his principle with the mechanical theory. Professor W. Thomson gave an account of Carnot's theory (Trans. Roy. Soc. Edin., Jan. 1849), in which he recognized the discrepancy between Clapeyron's statement and Joule's experiments, but did not see his way out of the difficulty. He therefore adopted Carnot's principle provisionally, and proceeded to calculate a table of values of Carnot's function F't, from the values of the total-heat and vapour-pressure of steam then recently determined by Regnault (Mémoires de l'Institut de Paris, 1847). In making the calculation, he assumed that the specific volume v of saturated steam at any temperature T and pressure is that given by the gaseous laws, pv=RT. The results are otherwise correct so far as Regnault's data are accurate, because the values of the efficiency per degree F't are not affected by any assumption with regard to the nature of heat. He obtained the values of the efficiency F't over a finite range from to o° C., by adding up the values of F't for the separate degrees. This latter proceeding is inconsistent with the mechanical theory, but is the 20. Combination of Carnot's Principle with the Mechanicalp Theory. A very instructive paper, as showing the state of the science of heat about this time, is that of C. H. A. Holtzmann, "On the Heat and Elasticity of Gases and Vapours "(Mannheim, 1845; Taylor's Scientific Memoirs, iv. 189). He points out that the theory of Laplace and Poisson does not agree with facts when applied to vapours, and that Clapeyron's formulae, 1 correct method on the assumption that the heat given up to the condenser is equal to that taken from the source. The values he obtained for F't agreed very well with those previously given by Carnot and Clapeyron, and showed that this function diminishes with rise of temperature roughly in the inverse ratio of T, as suggested by Joule. Professor W. Thomson (Lord Kelvin) in a paper "L On the Dynamical Theory of Heat" (Trans. Roy. Soc. Edin., 1851, first published in the Phil. Mag., 1852) gave a very clear statement of the position of the theory at that time. He showed that the value F-J/T, assumed for Carnot's function by Clausius without any experimental justification, rested solely on the evidence of Joule's experiment, and right possibly not be true at all temperatures. Assuming the value J/T with this reservation, he gave as the expression for the efficiency over a finite range h to lo C., or T1 to To Abs., the result, W/H=(4-4)/(4+273) = (T1−To)/T, (4) the principal results were detailed. Assuming the value of Joule's equivalent, Rankine deduced the value 0-2404 for the specific heat of air at constant pressure, in place of 0.267 as found by Delaroche and Bérard. The subsequent verification of this value by Regnault (Comptes rendus, 1853) afforded strong confirmation of the accuracy of Joule's work. In a note appended R. J. E. Clausius (Pogg. Ann., 1850, 79, p. 369) and W. J. M. to the abstract in the Phil. Mag. Rankine states that he has Rankine (Trans. Roy. Soc. Edin., 1850) were the first to develop succeeded in proving that the maximum efficiency of an engine the correct equations of thermodynamics on the mechanical working in a Carnot cycle of finite range to to is of the form theory. When heat was supplied to a body to change its tempera-(4-to)/(h−k), where k is a constant, the same for all substances. ture or state, part remained in the body as intrinsic heat energy This is correct if represents temperature Centigrade, and E, but part was converted into external work of expansion Wk=-273. and ceased to exist as heat. The part remaining in the body was always the same for the same change of state, however performed, as required by Carnot's fundamental axiom, but the part corresponding to the external work was necessarily different for different values of the work done. Thus in any cycle in which the body was exactly restored to its initial state, the heat remaining in the body would always be the same, or as Carnot puts it, the quantities of heat absorbed and given out in its diverse transformations are exactly "compensated," so far as the body is concerned. But the quantities of heat absorbed and given out are not necessarily equal. On the contrary, they differ by the equivalent of the external work done in the cycle. Applying this principle to the case of steam, Clausius deduced a fact previously unknown, that the specific heat of steam maintained in a state of saturation is negative, which was also deduced by Rankine (loc. cit.) about the same time. In applying the principle to gases Clausius assumes (with Mayer and Holtzmann) that the heat absorbed by a gas in isothermal expansion is equivalent to the work done, but he does not appear to be acquainted with Joule's experiment, and the reasons he adduces in support of this assumption are not conclusive. This being admitted, he deduces from the energy principle alone the propositions already given by Carnot with reference to gases, and shows in addition that the specific heat of a perfect gas must be independent of the density. In the second part of his paper he introduces Carnot's principle, which he quotes as follows: "The performance of work is equivalent to a transference of heat from a hot to a cold body without the quantity of heat being thereby diminished." This is not Carnot's way of stating his principle (see § 15), but has the effect of exaggerating the importance of Clapeyron's unnecessary assumption. By equating the expressions given by Carnot for the work done and the heat absorbed in the expansion of a gas, he deduces (following Holtzmann) the value J/T for Carnot's function F't (which Clapeyron denotes by 1/C). He shows that this assumption gives values of Carnot's function which agree fairly well with those calculated by Clapeyron and Thomson, and that it leads to values of the mechanical equivalent not differing greatly from those of Joule. Substituting the value J/T for C in the analytical expressions given by Clapeyron for the latent heat of expansion and vaporization, these relations are immediately reduced to their modern form (see THERMODYNAMICS, § 4). Being unacquainted with Carnot's original work, but recognizing the invalidity of Clapeyron's description of Carnot's cycle, Clausius substituted a proof consistent with the mechanical theory, which he based on the axiom that "heat cannot of itself pass from cold to hot." The proof on this basis involves the application of the energy principle, which does not appear to be necessary, and the axiom to which final appeal is made does not appear more convincing than Carnot's. Strange to say, Clausius did not in this paper give the expression for the efficiency in a Carnot cycle of finite range (Carnot's Ft) which follows immediately from the value J/T assumed for the efficiency F't of a cycle of infinitesimal range at the temperature / C or T Abs.. which, he observed, agrees in form with that found by Rankine. 22. Availability of Heat of Combustion.-Taking the value 1.13 kilogrammetres per kilo-calorie for 1° C. fall of temperature at 100° C., Carnot attempted to estimate the possible performance of a steam-engine receiving heat at 160° C. and rejecting it at 40° C. Assuming the performance to be simply proportional to the temperature fall, the work done for 120° fall would be 134 kilogrammetres per kilo-calorie. To make an accurate calculation required a knowledge of the variation of the function F't with temperature. Taking the accurate formula of § 20, the work obtainable is 118 kilogrammetres per kilo-calorie, which is 28% of 426, the mechanical equivalent of the kilo-calorie in kilogrammetres. Carnot pointed out that the fall of 120° C. utilized in the steam-engine was only a small fraction of the whole temperature fall obtainable by combustion, and made an estimate of the total power available if the whole fall could be utilized, allowing for the probable diminution of the function F't with rise of temperature. His estimate was 3.9 million kilogrammetres per kilogramme of coal. This was certainly an over-estimate, but was surprisingly close, considering the scanty data at his disposal. it directly by internal combustion. This avoids the limitation imposed by the use of a separate boiler, which as we have seen reduces the possible efficiency at least 50%. Even with internal combustion, however, the full range of temperature is not available, because the heat cannot conveniently in practice be communicated to the working fluid at constant temperature, owing to the large range of expansion at constant temperature required for the absorption of a sufficient quantity of heat. Air-engines of this type, such as Stirling's or Ericsson's, taking in heat at constant temperature, though theoretically the most In reality the fraction of the heat of combustion available, perfect, are bulky and mechanically inefficient. In practical even in an ideal engine and apart from practical limitations, is engines the heat is generated by the combustion of an explosive much less than might be inferred from the efficiency formula of mixture at constant volume or at constant pressure. The heat the Carnot cycle. In applying this formula to estimate the is not all communicated at the highest temperature, but over availability of the heat it is usual to take the temperature a range of temperature from that of the mixture at the beginning obtainable by the combustion of the fuel as the upper limit of of combustion to the maximum temperature. The earliest temperature in the formula. For carbon burnt in air at constant instance of this type of engine is the lycopodium engine of pressure without any loss of heat, the products of combustion M.M. Niepce, discussed by Carnot, in which a combustible might be raised 2300° C. in temperature, assuming that the mixture of air and lycopodium powder at atmospheric pressure specific heats of the products were constant and that there was was ignited in a cylinder, and did work on a piston. The no dissociation. If all the heat could be supplied to the working early gas-engines of E. Lenoir (1860) and N. Otto and E. fluid at this temperature, that of the condenser being 40° C., Langen (1866), operated in a similar manner with illuminating the possible efficiency by the formula of § 20 would be 89%. gas in place of lycopodium. Combustion in this case is effected But the combustion obviously cannot maintain so high a tem- practically at constant volume, and the maximum efficiency perature if heat is being continuously abstracted by a boiler. theoretically obtainable is 1-logy/(r-1), where r is the ratio Suppose that is the maximum temperature of combustion as of the maximum temperature to the initial temperature →o. above estimated, 0" the temperature of the boiler, and that In order to obtain this efficiency it would be necessary to follow of the condenser. Of the whole heat supplied by combustion Carnot's rule, and expand the gas after ignition without loss represented by the rise of temperature -0°, the fraction or gain of heat from ' down to 0°, and then to compress it (0′-0′′)/(0'-0°) is the maximum that could be supplied to the at to its initial volume. If the rise of temperature in com, boiler, the fraction (0′′−0o)/(0′ -0°) being carried away with the bustion were 2300° C., and the initial temperature were o° C. waste gases. Of the heat supplied to the boiler, the fraction or 273° Abs., the theoretical efficiency would be 73.3%, which (0-0)/0" might theoretically be converted into work. The is much greater than that obtainable with a boiler. But in problem in the case of an engine using a separate working fluid, order to reach this value, it would be necessary to expand the like a steam-engine, is to find what must be the temperature " mixture to about 270 times its initial volume, which is obviously of the boiler in order to obtain the largest possible fraction of the impracticable. Owing to incomplete expansion and rapid heat of combustion in the form of work. It is easy to show that 0" cooling of the heated gases by the large surface exposed, the must be the geometric mean of ' and 0o, or 0" = √0′0。. Taking actual efficiency of the Lenoir engine was less than 5%, and of '-0o 2300° C., and 0=313° Abs. as before, we find @"= the Otto and Langen, with more rapid expansion, about 10%. 903 Abs. or 630° C. The heat supplied to the boiler is then Carnot foresaw that in order to render an engine of this type 74-4% of the heat of combustion, and of this 65.3% is converted practically efficient, it would be necessary to compress the into work, giving a maximum possible efficiency of 49% in mixture before ignition. Compression is beneficial in three place of 89%. With the boiler at 160° C., the possible efficiency, ways: (1) it permits a greater range of expansion after ignition; calculated in a similar manner, would be 26-3%, which shows (2) it raises the mean effective pressure, and thus improves the that the possible increase of efficiency by increasing the tem- mechanical efficiency and the power in proportion to size and perature range is not so great as is usually supposed. If the weight; (3) it reduces the loss of heat during ignition by reducing temperature of the boiler were raised to 300° C., corresponding the surface exposed to the hot gases. In the modern gas or to a pressure of 1260 lb per sq. in., which is occasionally surpassed petrol motor, compression is employed as in Carnot's cycle, in modern flash-boilers, the possible efficiency would be 40%. but the efficiency attainable is limited not so much by consideraThe waste heat from the boiler, supposed perfectly efficient, tions of temperature as by limitations of volume. It is impracticwould be in this case 11%, of which less than a quarter could able before combustion at constant volume to compress a rich be utilized in the form of work. Carnot foresaw that in order mixture to much less than th of its initial volume, and, for to utilize a larger percentage of the heat of combustion it would mechanical simplicity, the range of expansion is made equal be necessary to employ a series of working fluids, the waste heat to that of compression. The cycle employed was patented from one boiler and condenser serving to supply the next in the in 1862 by Beau de Rochas (d. 1892), but was first successfully series. This has actually been effected in a few cases, e.g. carried out by Otto (1876). It differs from the Carnot cycle steam and SO2, when special circumstances exist to compensate in employing reception and rejection of heat at constant volume for the extra complication. Improvements in the steam-engine instead of at constant temperature. This cycle is not so efficient since Carnot's time have been mainly in the direction of reducing as the Carnot cycle for given limits of temperature, but, for the waste due to condensation and leakage by multiple expansion, given limits of volume imposed, it gives a much higher efficiency superheating, &c. The gain by increased temperature range than the Carnot cycle. The efficiency depends only on the has been comparatively small owing to limitations of pressure, range of temperature in expansion and compression, and is and the best modern steam-engines do not utilize more than 20% given by the formula (0'-0")/0', where ' is the maximum of the heat of combustion. This is in reality a very respectable temperature, and " the temperature at the end of expansion. fraction of the ideal limit of 40% above calculated on the The formula is the same as that for the Carnot cycle with the assumption of 1260 lb initial pressure, with a perfectly efficient same range of temperature in expansion. The ratio 0'0" is boiler and complete expansion, and with an ideal engine which, where r is the given ratio of expansion or compression, does not waste available motive power by complete condensation of the steam before it is returned to the boiler. 23. Advantages of Internal Combustion.-As Carnot pointed out, the chief advantage of using atmospheric air as a working fluid in a heat-engine lies in the possibility of imparting heat to and y is the ratio of the specific heats of the working fluid. Assuming the working fluid to be a perfect gas with the same properties as air, we should have y 1.41. Taking r5, the formula gives 48% for the maximum possible efficiency. The actual products of combustion vary with the nature of the fuel employed, and have different properties from air, but the | realize as much as 34% indicated efficiency, which is 90% of efficiency is found to vary with compression in the same manner the maximum possible, showing how perfectly all avoidable heat as for air. For this reason a committee of the Institution of Civil losses have been minimized. Engineers in 1905 recommended the adoption of the air-standard for estimating the effects of varying the compression ratio, and defined the relative efficiency of an internal combustion engine as the ratio of its observed efficiency to that of a perfect air-engine with the same compression. 24. Effect of Dissociation, and Increase of Specific Heat.-One of the most important effects of heat is the decomposition or dissociation of compound molecules. Just as the molecules of a vapour combine with evolution of heat to form the more complicated molecules of the liquid, and as the liquid molecules require the addition of heat to effect their separation into molecules of vapour; so in the case of molecules of different kinds which combine with evolution of heat, the reversal of the process can be effected either by the agency of heat, or indirectly by supplying the requisite amount of energy by electrical or other methods. Just as the latent heat of vaporization diminishes with rise of temperature, and the pressure of the dissociated vapour molecules increases, so in the case of compound molecules in general the heat of combination diminishes with rise of temperature, and the pressure of the products of dissociation increases. There is evidence that the compound carbon dioxide, CO2, is partly dissociated into carbon monoxide and oxygen at high temperatures, and that the proportion dissociated increases with rise of temperature. There is a very close analogy between these phenomena and the vaporization of a liquid. The laws which govern dissociation are the same fundamental laws of thermodynamics, but the relations involved are necessarily more complex on account of the presence of different kinds of molecules, and present special difficulties for accurate investigation in the case where dissociation does not begin to be appreciable until a high temperature is reached. It is easy, however, to see that the general effect of dissociation must be to diminish the available temperature of combustion, and all experiments go to show that in ordinary combustible mixtures the rise of temperature actually attained is much less than that calculated as in § 22, on the assumption that the whole heat of combustion is developed and communicated to products of constant specific heat. The defect of temperature observed can be represented by supposing that the specific heat of the products of combustion increases with rise of temperature. This is the case for CO2 even at ordinary temperatures, according to Regnault, and probably also for air and steam at higher temperatures. Increase of specific heat is a necessary accompaniment of dissociation, and from some points of view may be regarded as merely another way of stating the facts. It is the most convenient method to adopt in the case of products of combustion consisting of a mixture of CO2 and steam with a large excess of inert gases, because the relations of equilibrium of dissociated molecules of so many different kinds would be too complex to permit of any other method of expression. It appears from the researches of Dugald Clerk, H, le Chatelier and others that the apparent specific heat of the products of combustion in a gas-engine may be taken as approximately 34 to 33 in place of 24 at working temperatures between 1000° C. and 1700° C., and that the ratio of the specific heats is about 1.29 in place of 141. This limits the availability of the heat of combustion by reducing the rise of temperature actually obtainable in combustion at constant volume by 30 or 40%, and also by reducing the range of temperature 0'0" for a given ratio of expansion from " to 29. The formula given in § 21 is no longer quite exact, because the ratio of the specific heats of the mixture during compression is not the same as that of the products of combustion during expansion. But since the work done depends principally on the expansion curve, the ratio of the range of temperature in expansion ('-0") to the maximum temperature ' will still give a very good approximation to the possible efficiency. Taking =5, as before, for the compression ratio, the possible efficiency is reduced from 48% to 38%, if y=1.29 instead of 1.41. A large gas-engine of the present day with 75 may actually It is often urged that the gas-engine is relatively less efficient than the steam-engine, because, although it has a much higher absolute efficiency, it does not utilize so large a fraction of its temperature range, reckoning that of the steam-engine from the temperature of the boiler to that of the condenser, and that of the gas-engine from the maximum temperature of combustion to that of the air. This is not quite fair, and has given rise to the mistaken notion that "there is an immense margin for improvement in the gas-engine," which is not the case if the practical limitations of volume are rightly considered. If expansion could be carried out in accordance with Carnot's principle of maximum efficiency, down to the lower limit of temperature 0o, with rejection of heat at 0, during compression to the original volume do, it would no doubt be possible to obtain an ideal efficiency of nearly 80%. But this would be quite impracticable, as it would require expansion to about 100 times vo, or 500 times the compression volume. Some advantage no doubt might be obtained by carrying the expansion beyond the original volume. This has been done, but is not found to be worth the extra complication. A more practical method, which has been applied by Diesel for liquid fuel, is to introduce the fuel at the end of compression, and adjust the supply in such a manner as to give combustion at nearly constant pressure. This makes it possible to employ higher compression, with a corresponding increase in the ratio of expansion and the theoretical efficiency. With a compression ratio of 14, an indicated efficiency of 40% has been obtained in this way, but owing to additional complications the brake efficiency was only 31%, which is hardly any improvement on the brake efficiency of 30% obtained with the ordinary type of gas-engine. Although Carnot's principle makes it possible to calculate in every case what the limiting possible efficiency. would be for any kind of cycle if all heat losses were abolished, it is very necessary, in applying the principle to practical cases, to take account of the possibility of avoiding the heat losses which are supposed to be absent, and of other practical limitations in the working of the actual engine. An immense amount of time and ingenuity has been wasted in striving to realize impossible margins of ideal efficiency, which a close study of the practical conditions would have shown to be illusory. As Carnot remarks at the conclusion of his essay: Economy of fuel is only one of the conditions a heat-engine must satisfy; in many cases it is only secondary, and must often give way to considerations of safety, strength and wearing qualities of the machine, of smallness of space occupied, or of expense in erecting. To know how to appreciate justly in each case the considerations of convenience and economy, to be able to distinguish the essential from the accessory, to balance all fairly, and finally to arrive at the best result by the simplest means, such must be the principal talent of the man called on to direct and co-ordinate the work of his fellows for the attainment of a useful object of any kind." TRANSFERENCE OF HEAT 25. Modes of Transference. There are three principal modes of transference of heat, namely (1) convection, (2) conduction, and (3) radiation. (1) In convection, heat is carried or conveyed by the motion of heated masses of matter. The most familiar illustrations of this method of transference are the heating of buildings by the circulation of steam or hot water, or the equalization of temperature of a mass of unequally heated liquid or gas by convection currents, produced by natural changes of density or by artificial stirring. (2) In conduction, heat is transferred by contact between contiguous particles of matter and is passed on from one particle to the next without visible relative motion of the parts of the body. A familiar illustration of conduction is the passage of heat through the metal plates of a boiler from the fire to the water inside, or the transference of heat from a soldering bolt to the solder and the metal with which it is placed in contact. (3) In radiation, the heated body gives rise to a motion of | and also by convection and conduction to the surrounding air, vibration in the aether, which is propagated equally in all increases much more rapidly than in simple proportion to the directions, and is reconverted into heat when it encounters any temperature difference, and the rate of increase of each follows obstacle capable of absorbing it. Thus radiation differs from a different law. At a later date Sir John Herschel measured the conduction and convection in taking place most perfectly in the intensity of the solar radiation at the surface of the earth, and absence of matter, whereas conduction and convection require endeavoured to form an estimate of the temperature of the sun material communication between the bodies concerned. by comparison with terrestrial sources on the assumption that the intensity of radiation was simply proportional to the temperature difference. He thus arrived at an estimate of several million degrees, which we now know would be about a thousand times too great. The application of Newton's law necessarily leads to absurd results when the difference of temperature is very large, but the error will not in general exceed 2 to 3% if the temperature difference does not exceed 10° C., and the percentage error is proportionately much smaller for smaller differences. In the majority of cases of transference of heat all three modes of transference are simultaneously operative in a greater or less degree, and the combined effect is generally of great complexity. The different modes of transference are subject to widely different laws, and the difficulty of disentangling their effects and subjecting them to calculation is often one of the most serious obstacles in the experimental investigation of heat. In space void of matter, we should have pure radiation, but it is difficult to obtain so perfect a vacuum that the effects of the residual gas in transferring heat by conduction or convection are inappreciable. In the interior of an opaque solid we should have pure conduction, but if the solid is sensibly transparent in thin layers there must also be an internal radiation, while in a liquid or a gas it is very difficult to eliminate the effects of convection. These difficulties are well illustrated in the historical development of the subject by the experimental investigations which have been made to determine the laws of heat-transference, such as the laws of cooling, of radiation and of conduction. 26. Newton's Law of Cooling.-There is one essential condition common to all three modes of heat-transference, namely, that they depend on difference of temperature, that the direction of the transfer of heat is always from hot to cold, and that the rate of transference is, for small differences, directly proportional to the difference of temperature. Without difference of temperature there is no transfer of heat. When two bodies have been brought to the same temperature by conduction, they are also in equilibrium as regards radiation, and vice versa. If this were not the case, there could be no equilibrium of heat defined by equality of temperature. A hot body placed in an enclosure of lower temperature, e.g. a calorimeter in its containing vessel, generally loses heat by all three modes simultaneously in different degrees. The loss by each mode will depend in different ways on the form, extent and nature of its surface and on that of the enclosure, on the manner in which it is supported, on its relative position and distance from the enclosure, and on the nature of the intervening medium. But provided that the difference of temperature is small, the rate of loss of heat by all modes will be approximately proportional to the difference of temperature, the other conditions remaining constant. The rate of cooling or the rate of fall of temperature will also be nearly proportional to the rate of loss of heat, if the specific heat of the cooling body is constant, or the rate of cooling at any moment will be proportional to the difference of temperature. This simple relation is commonly known as Newton's law of cooling, but is limited in its application to comparatively simple cases such as the foregoing. Newton himself applied it to estimate the temperature of a red-hot iron ball, by observing the time which it took to cool from a red heat to a known temperature, and comparing this with the time taken to cool through a known range at ordinary temperatures. According to this law if the excess of temperature of the body above its surroundings is observed at equal intervals of time, the observed values will form a geometrical progression with a common ratio. Supposing, for instance, that the surrounding temperature were o° C., that the red-hot ball took 25 minutes to cool from its original temperature to 20° C., and 5 minutes to cool from 20° C. to 10° C., the original temperature is easily calculated on the assumption that the excess of temperature above o° C. falls to half its value in each interval of 5 minutes. Doubling the value 20° at 25 minutes five times, we arrive at 640° C. as the original temperature. No other method of estimation of such temperatures was available in the time of Newton, but, as we now know, the simple law of proportionality to the temperature difference is inapplicable over such large ranges of temperature. The rate of loss of heat by radiation, 27. Dulong and Petit's Empirical Laws of Cooling.-One of the most elaborate experimental investigations of the law of cooling was that of Dulong and Petit (Ann. Chim. Phys., 1817, 7, pp. 225 and 337), who observed the rate of cooling of a mercury thermometer from 300° C. in a water-jacketed enclosure at various temperatures from o° C. to 80° C. In order to obtain the rate of cooling by radiation alone, they exhausted the enclosure as perfectly as possible after the introduction of the thermometer, but with the imperfect appliances available at that time they were not able to obtain a vacuum better than about 3 or 4 mm. of mercury. They found that the velocity of cooling V in a vacuum could be represented by a formula of the type in which is the temperature of the thermometer, and to that of the enclosure, a is a constant having the value 1.0075, and the coefficient A depends on the form of the bulb and the nature of its surface. For the ranges of temperature they employed, this formula gives much better results than Newton's, but it must be remembered that the temperatures were expressed on the arbitrary scale of the mercury thermometer, and were not corrected for the large and uncertain errors of stem-exposure (sec THERMOMETRY). Moreover, although the effects of cooling by convection currents afe practically eliminated by exhausting to 3 or 4 mm. (since the density of the gas is reduced to 1/200th while its viscosity is not appreciably affected), the rate of cooling by conduction is not materially diminished, since the conductivity, like the viscosity, is nearly independent of pressure. It has since been shown by Sir William Crookes (Proc. Roy. Soc., 1881, 21, p. 239) that the rate of cooling of a mercury thermometer in a vacuum suffers a very great diminution when the pressure is reduced from 1 mm. to 001 mm., at which pressure the effect of conduction by the residual gas has practically disappeared. Dulong and Petit also observed the rate of cooling under the same conditions with the enclosure filled with various gases. They found that the cooling effect of the gas could be represented by adding to the term already given as representing radiation, an expression of the form They found that the cooling effect of convection, unlike that of radiation, was independent of the nature of the surface of the thermometer, whether silvered or blackened, that it varied as some power c of the pressure p, and that it was independent of the absolute temperature of the enclosure, but varied as the excess temperature (-lo) raised to the power 1.233. This highly artificial result undoubtedly contains some clements of truth, but could only be applied to experiments similar to those from which it was derived. F. Hervé de la Provostaye and P. Q. Desains (Ann. Chim. Phys., 1846, 16, p. 337), in repeating these experiments under various conditions, found that the coefficients A and B were to some extent dependent on the temperature, and that the manner in which the cooling effect varied with the pressure depended on the form and size of the enclosure. It is evident that this should be the case, since the cooling effect of the gas depends partly on convective currents, |