well adopt this equation as the true expression of the relations between p, u, and t. Again, the values of u, as stated in the 7th column, are not those calcu lated in the way above given, but those very nearly the same, which are obtained by a division of the values of Apu, given in column 10, by the corresponding values of A p. Let us now briefly consider the signification of the magnitudes represented in the table. Q, as already said, is the total quantity of heat which is expended in order first to heat one kilogram of water at 0° to t°, and then to convert the water of to, under the corresponding pressure p, into saturated vapor of to. A part of this quantity Q, namely, Apu, is spent in external work; it is, therefore, no longer contained in the saturated vapor at t°; the total quantity of heat contained in the vapor (excepting that already in the water at 0°) is only J-Q-Apu. The values of J are represented in the 11th column of our table. In order to transform water at to into saturated vapor at to, the quantity of heat is necessary, whose values are exhibited in the 6th vertical series of the table. It is this magnitude r which is usually designated as latent heat; an incorrect expression, however, if we mean thereby to indicate the quantity of heat employed in abolishing the cohesion of the particles of water, and hence spent in an external work, for a part of the quantity r, namely, A pu, is consumed by that external work. Only the remainder, p=r-Apu, can be regarded as the internal latent heat of vapor, while the magnitude r might, according to Clausius, be designated as evaporation-heat. It is evident that for a correct calculation of the effects of steam-engines the values of v must be taken into account as they are set forth in our table, and not hose reckoned after M. Gay-Lussac's law. It is to be observed, however, tn a comparison of the values of V and v, given in two preceding pages, that in the last the volume of one gram of vapor is expressed in cubic centimetres, iwhile in the first it is stated in cubic metres. From a consideration of the numbers grouped together in the table it will be seen that of the quantity of heat conveyed to the water in the boiler only a very small part is expended in mechanical work; for the quantity Apu employed in such work is but an inconsiderable fraction even of the evaporation-heat r, about for 100°, and for 160°. The internal latent va or-heat p abides with the vapor at its exit from the machine, and hence can do no work. This quantity of heat p can only be in part regained. ρ This circumstance occasioned the constructing of power machines, in which the elasticity of heated air might operate instead of steam. Such machines, constructed particularly after Ericsson's designs, and known by the name of caloric engines, have been repeatedly introduced into practice with high expectations, but have been as often abandoned because their performance fell far short of that of the steam-engine. VIII.-ACTION OF SATURATED STEAM DURING EXPANSION. The results thus far obtained enable us to form a correct idea of the action of steam in our expansion steam-engines. As we cannot, however, here develop the equations necessary for the general solution of this problem, we must content ourselves with the consideration of special cases. If, under the piston of a steam-cylinder, there be just one kilogram of saturated steam of 160° C., we find for this steam by our table, v=0.3002 cubic metre. Let the steam be now supposed slowly to expand, and the pressure on the piston to be, at each moment, equal to the corresponding tension of the steam. During this expansion both the temperature and elasticity of the steam is lowered. Suppose the steam to expand till its temperature has fallen 5°, and hence in the case under consideration from 160° to 155°. We will inquire now how much heat must be supplied to this expanding steam, if, during the expansion in question, no condensation take place and the quantity of steam is to remain unaltered. By our table we have for 155°, v0.3387 cubic metre. The volume of the steam has therefore increased by V=v'—v=0.0385 of a cubic metre. The work done during this expansion we know at least approxip+p' mately, L=V. whence, in our special case, we put 2 L=0.0385 59415.7-2287.5 metre-kilograms; the quantity of heat requisite for this work is, At the beginning of the expansion the total heat contained in the steam, J=610.53, at the termination of the expansion, J'=609.35; thus we see that during the expansion, J-J', equal in units of heat to 1.18, has disappeared from the steam. But this quantity of heat is not sufficient to execute the work amounting to 2287.5 metre-kilograms; so that 5.39-1.18-4.21 units of heat must be added from without, if the steam is to expand in the manner above stated, without diminution of the quantity of steam. If we repeat the same process for the temperature standing in the beginning at 120°, (instead of 160°, as in the preceding case,) we find, V=v-v=0.1433. p+p' 18767.2. A L=6.34 units of heat. J-J' 1.12 units of heat. In this last case, therefore, an addition of 6.34-1.12-5.22 units of heat is needed. Now, if the numerical values just calculated, make no pretension to exactness, they still serve to show that a considerable addition of heat is necessary if the steam is to expand in the way specified, without the occurrence of partial condensation. But since, in our expansion steam-engines, .no further addition of heat ensues after the shutting off of the supply of steam, it is clear that in consequence of the expansion a partial condensation of steam must follow. The last part, therefore, of the proposition announced by Pambour, "Steam, while expanding without heat being supplied, remains saturated, and no vapor is thereby precipitated," is inadmissible; much rather would it be proper thus to modify the proposition, "While steam is expanding without a supply of heat, it remains, indeed, saturated, but thereby is a proportional quantity of vapor precipitated." Hence, at the end of the expansion the quantity of steam is less than at its commencement. It is through the condensation of steam that the heat must be furnished, which is in deficiency, for the performance of the work of expansion. This important discovery, respecting the action of steam during its expansion, was made almost simultaneously by Clausius and Rankine. It is clear that the theory of steam-engines must, from this fact, undergo an essential modification. That the expansion of steam is attended by partial condensation, admits of being likewise experimentally demonstrated. Into a glass balloon, (Fig. 12,) Fig. 12. which is wrapped around with wire and provided with a brass appendage which contains a duct, closed at will by a cock, we introduce a little water, so that the glass may be moistened within by a few drops. The balloon is then screwed to a compressing pump, and so much air is pumped in that the tension in the interior of the balloon shall amount to about three atmospheres. If the cock be now closed, and the balloon be unscrewed and laid in some warm place, whether in the vicinity of a stove or in the sunshine, the vapors in the interior will, after a while, have acquired the elasticity corresponding to the temperature, and appear perfectly transparent. If the cock be now opened, air and vapor will rapidly escape from the balloon, and the latter will be filled with a thick mist. In a somewhat altered form, the experiment admits of being executed as follows: Let the bell of an air-pump be sprinkled on the inside with a little water, and after being placed on the plate of the pump, be left some time in a rather warm chamber. As soon now as we begin to discharge the air, a white mist will be formed within the bell. A corresponding experiment has been instituted by Hirn on a large scale. A straight copper cylinder, two metres long and 15 centimetres in diameter, was closed at both ends by flat plates, in the middle of which were openings two centimetres wide and closed by plates of glass cemented therein. This cylinder was on one side placed in connection with a steam-boiler, while on the other it bore a discharge tube which was furnished with a wide cock. Into the cylinder was now introduced steam of a high tension, while the escape-cock was only partially opened, so that all the air might find an issue. The highly condensed vapor, (five atmospheres, for example,) which in this way fills the cylinder, is now perfectly transparent, so that all objects are plainly visible through the glass plates mentioned above. If the influx of steam from the boiler be afterwards wholly shut off, and the escape-cock be suddenly and fully opened, so that the steam promptly expands to a tension of one atmosphere, a mist of such density is formed in the cylinder that its contents. appear completely opaque. IX. THE MELTING OF ICE. After having applied the principles of the mechanical theory of heat to the formation of steam, we will proceed to consider them in relation to the phenomena of melting, and shall here treat exclusively of the melting of ice. Let us suppose one kilogram of ice at 0° C. to be contained in a vessel, under the pressure of one atmosphere. If heat be communicated to this ice, while the pressure remains unaltered, the ice passes gradually into water, but the temperature continues at 0° until the liquefaction is wholly completed. From this moment first begins the elevation of temperature, if the supply of heat is maintained. The quantity of heat which is necessary thus, under the pressure of one atmosphere, to convert one kilogram of ice at 0° into water at 0°, taking the mean of the best experiments, is, r=79.035 units of heat. This quantity is usually called, as in the case of steam, the latent heat. The phenomenon of melting is wholly analogous to that of evaporation. A part of the heat supplied to the ice is expended in overcoming the cohesion, the other part in external work, as, during the melting, a change of volume takes place. As in the formation of steam, if it takes place under the conditions set forth in § 7 the temperature at which the transformation into vapor occurs is a function of the pressure exerted on the piston, so that we might expect that the temperature of the melting ice depends also on the pressure under which it stands; that the temperature of fusion also varies with the pressure. From the analogy between vaporization and fusion, we are authorized to apply to the phenomenon of melting the equations developed in section VII, and therefore the second leading equation of the mechanical theory of heat u = A.T. But while, in the formation of steam, the magnitude was given through the experiment, it is here wholly unknown; on the other hand, we know the quantities r, A, T, and u, and hence have As we have above seen, r=79.035. 1 424 (111) The absolute temperature at which, under the pressure of the atmosphere, the melting of ice takes place, is T=273°. We know, moreover, that A= For the computation of there is wanting, therefore, only the value of u. In the melting of ice, we know a diminution of volume takes place. The volume of one kilogram of water at 0° is v=0.001 cubic metre. The volume of one kilogram of ice at 0°, taking the mean of different computations, is w=0.00109; hence u=v-w=-0.00009, and therefore negative. If we place now for r, A, T, and u, in equation, (111,) the numerical values cited, the result is Should the difference of pressure, which corresponds to a lowering of temperature of 1°, be expressed, not in kilograms, but in atmospheres, we shall have to divide by 10333, and we then obtain that is to say, an augmentation of pressure from 1 to 132 atmospheres would correspond to a lowering of the melting point by 1° C.; an augmentation of the pressure by one atmosphere will therefore be followed by a lowering of the melting temperature of ice equal to 13, or 0.0075° C. That the temperature at which ice melts varies with the pressure, and that thus an elevation of the pressure corresponds to a lowering of the freezing point, was first theoretically demonstrated by James Thomson, (Proceedings of the Royal Society of Edinburgh,) and then by Clausius, (Pogg. Annal., LXXXI,) and was experimentally verified by William Thomson, (Pogg. Annal., LXXXI.) The latter availed himself, for this experiment, of a thermometer in which ether was employed instead of quicksilver as the thermometric fluid. The reservoir of this thermometer was three and a half inches long and three-eighths inch wide. On the tube, six and a half inches long, was a scale of five and a half inches length, divided into 220 equal parts. The extent of this scale corresponded to a difference of temperature of about 3° F., so that a division represented on an average of a degree of F. The thermometer was so regulated that it showed the temperature between 31° and 34° F. In order that the reservoir might not be compressed when submitted to a strong pressure, it was hermetically inclosed 1 in a wider glass tube; this outer tube contained enough quicksilver for the reservoir of the ether thermometer to be wholly surrounded by it. This thermometer was now plunged, together with a cylindrical tube filled with air, into Oersted's compressing apparatus, which was filled partly with water, partly with pieces of pure ice. By means of a ring of lead, care was taken to keep the water of the compressing vessel free from ice at that part of the thermometer on which the readings were to be made. A pressure of 8.1, and again of 16.8 atmospheres, produced a sinking of the thermometer by 7 and 16 of the divisions of the scale, and thus by. =0.106° F., and 16.5 71 7.5 =0.232° F.; which very nearly coincides with the theoretically calculated depression of 0.109° and 0.227° F. Fig. 13. Fig. 14. From the above developments and observations it might be expected that water under very high pressure must remain fluid at relatively low temperatures. That this, indeed, is the case, is confirmed by the experiments long since conducted by Williams in Quebec, in order to measure the force with which freezing water expands. He exposed to intense cold thick iron bomb-shells, filled with water and closed by means of an iron plug firmly driven in. At a very low temperature the stopper was either driven out and then an icicle was projected from the opening, (Fig. 13,) or the bomb was burst, and in that case a sheet of ice protruded from the fissure, (Fig. 14.) The form of these extruded pieces of ice indicated conclusively that the water at a very low temperature still remained fluid, and was first converted into ice at the moment when it gained additional space. Fig. 15. In fine, Mousson has shown (Pogg. Annal., cv) that at a very low temperature ice may be rendered fluid by great pressure. The apparatus of which he availed himself for this is represented in section in Fig. 15, and on a smaller scale in side elevation in Fig. 16. Through the axis of a massive prism A of the best steel, four-cornered below and furnished above with the worm of a screw, a cylindrical cavity, 7.12 millimetres wide, is drilled, which, in its upper part, widens from b to a, in a slightly conical form, so that the mouth at a has a diameter of 8.61 millimetres. From above is driven into the cylindrical cavity a piece of pure copper g, somewhat conical at first, and fitting into the cavity a b so as to form above a perfect closure of the same. To the copper cylinder g is affixed a steel prolongation D, of like diameter with the cavity b c, and which, by application of the female screw E, can be pressed downwards so as to drive the copper cylinder g further into the cavity bc. Underneath the cylindrical cavity be is also a conical but rapidly widening cavity, into which fits the copper cone f, which, by means of the steel screw C, can be firmly pressed into its cavity. In order to perform the experiment, the screw C and the copper conef were first removed, the whole apparatus was inverted, so that E was below, A above, and the free part of the cavity be, above g, was filled with water that had been boiled; the copper index d was now lowered into this water. With the position unchanged, while d thus stood upon g, the whole apparatus was exposed to a low tempera |