and therefore so are their respective areas between the same parallels. But another known property is, that, in the hyperbola aBc, area cKHa: BIHa:: log. GK-log. GH : log. GI— log. GH:: 4:3; as is evident by taking the logarithms of the last equation but one, and stating them in a proportion. Hence, area BIHa=3(cKHa) =ƒKHd; proving, conversely, that hyperbolas possess both the properties specified in the proposition. By omitting area EIHd, there remains EIKƒ— EBad; and, in the same way, EIKF = EBAD.

If Kc < KC, and Hd > HD, and if the hyperbolas do not again meet the other curves between AH and CK, we have area BIHA <BIHa. But EBAD = (BIHA), and EBad = (BIHa); wherefore, EBAD <EBad, and yet EIKF> EIKƒ; that is, two equals are respectively greater and, less than two equals, which is absurd. The like would follow, were Kc> KC and Hd < HD.

In however many points the hyperbolas might cut the other curves, yet three ordinates may always be taken so near to each other that the same absurdity will occur as before. For the middle ordinate BI, and either of the others, as AH or CK, may be drawn at pleasure, and the position of the third obtained from

It is, indeed, a supposable case, that the hyperbolas might touch the other curves in B and E; but these curves cannot be touched in every point by hyperbolas having all the same asymptotes. Suppose, however, the curve ABC were touched by two such hyperbolas; then, because the latter do not meet, some third hyperbola lying between them could cut the curve ABC in at least one point, B'; and because ABC and DEF cut the ordinates in the same ratio as the hyperbolas do, the curve DEF would be cut by its corresponding hyperbola in a point, E, situated in the same ordinate with B', Other two ordinates, therefore, being drawn near enough to B'E', would still produce the former absurdity. Hence, ABC and DEF can only be hyperbolas.

P

Let ABC (using the former figures) be a curve, such that whilst the temperature of a mass of air, under a constant pressure, undergoes any change indicated by HI on the scale of an air-thermometer, the corresponding change in its quantity of heat may be denoted by area BIHA, intercepted between the ordinates BI and A H perpendicular to HI; and let DEF be a similar sort of curve, making area EIH D≈ (BIHA), which, therefore, represents (art. 4) the change which the quantity of heat would have undergone had the air been confined in a close vessel during the change of temperature H I. Let the common scale of the air-thermometer, of which HI is a part, be continued downward to the point G, answering to -448° F.; then (art. 3) the volume had changed under a constant pressure in the ratio of G H to GI, whilst the temperature shifted from H to I. But now, suppose the air, which has acquired the temperature I, and undergone a change of heat BIHA, to be instantly restored to its original volume, whereby the temperature reaches the point K on the scale. It is evident, that the air is now brought to the same temperature as if its quantity of heat had undergone the change FK HD BIHA, with its original volume all the while invariable, i During the restoration of the air to its original volume its quantity of heat is supposed constant, but the density has been changed in the ratio of GH to G I, and the pressure (art. 1) in the same ratio compounded (art. 3) with the ratio of GI to GK for change of temperature; that is as GH to GK,

GI

GK

(GL)'= (GH)': But we had also BIHA FKHD; con

GH

sequently (art. 7) the curves A B C, D E F are hyperbolas*.

Since the variations of the area denote variations in the quantity of heat, whilst the corresponding variations of the abscissæ denote variations of temperature on the common scale, it follows, from the known property of the hyperbola, that whilst the variations of the quantity of heat in air under a constant pressure are uniform, the variations on the common scale of an air-thermometer form a geometrical progression. Or more generally, when the variations in the quantity of heat are uniform, those of the volume under a constant pressure form a geometrical progression; as do likewise the variations of pressure under a constant volume.

Having finished this investigation, it may be useful to give some account of the experiments alluded to in art. 4. I shall

*These curves would still have come out hyperbolas, if, throughout this investigation we had used, in place of 4 and 3, constant symbols, or any two numbers of which the first exceeds the second: so that the scale of the air-thermometer now investigated is not the consequence of a parti cular ratio between the areas of the two curves. It only requires a constant ratio. Now the constancy of this ratio (as will be noticed after) is strenuously contended for by Mr. Ivory and the French mathematicians, who, therefore, set out with the same data as I do. But in place of going through with the investigation of the scale of temperature, for which the above data, if true, are, as we have seen, amply sufficient, these great mathematicians stick fast in the middle of it, and assume, what cannot be proved, because directly at variance with the data,—that the common scale of the air-thermometer is the true one. Among many other absurdities, of which Mr. Ivory's new law affords a notable example, this assumption would require the above curves to be mere straight lines!-See Mécanique Celeste, tom. v., p. 128; Annales de Chimie, xxiii., 337; Phil. Mag., Nov. 1823, and Feb. 1827.

I flatter myself that the investigation now given of the scale of the air-thermometer, may be easily followed by such as are but little conversant in fluxions, provided they know a few of the simplest properties of the hyperbola. I have been the more sparing of fluxions, and have employed geometry for another reason; namely, because Mr. Ivory having egregiously mismanaged this problem when using fluxions, is anxious to darken the subject, and to shelter himself in pretending, that the problem is necessarily indeterminate or ambiguous in its result. Whoever attends to the above solution will readily see that, if once the data be admitted, there is no alternative as to the result.

first give a familiar illustration of the principle, and then state more nearly the actual mode of experimenting.

Suppose a close vessel to contain air of the same temperature and pressure as that in the apartment, and to be furnished with a gauge for showing any minute change of pressure, In this state, let it be carried into another room 4° warmer, when the gauge will soon indicate an increase of elasticity, which we may call 4°. If the vessel admit of being now opened sufficiently, and shut again so promptly, as just for the moment to allow the included air to regain the external pressure, without affording time for its abstracting heat from the vessel, the gauge will in a little time indicate an increase of 1°.

The reason of this may be traced, in a general way, to the known circumstance that air, of whatever density, is cooled by dilatation. The gauge, while indicating 4°, shews the included air to be denser than the external. On opening the vessel, therefore, the superior elastic force within expels a portion of air, leaving the remainder, of course, rarer, and consequently cooled, as the result shews, one degree below the temperature of the second room. At the moment of re-shutting the vessel the elasticity of the included air is in equilibrio with the external pressure; but it must then be more dense than the air of the second apartment, otherwise, its temperature being lower, it could not balance the external pressure. On regaining, therefore, the lost degree of temperature, the excess of density shows itself by the gauge. The escape of a small portion of air does not alter the result, because it only carries off its own heat.

Hence, the quantity of heat which had kept up the temperature of the air 4° above its original pitch while confined in the vessel, shows itself to be only able, under the original pressure, to maintain an excess of 3° above the original temperature. The quantity of heat, therefore, which would change the temperature of air one degree, and which is usually called its specific heat, is one-third greater when the air is free to change its volume under a constant pressure, than if confined in an inextensible vessel *.

By specific heat, some profess to mean the absolute or whole quantity

The experiment would have been essentially the same had the vessel been only closed under the temperature and pressure in the second apartment, and then air injected till the gauge stood at 4°, after the temperature had settled. The rest of the experiment, as to opening and shutting, being as before described. Now, in either method, it is obvious that, at the instant of re-shutting the vessel, the included air, with reference to pressure, had lost all the 4o; while, as the second indication of the gauge shews, it had only lost 3° with regard to density,one of the lost degrees of pressure being due to temporary depression of temperature. Hence, the height of the barometer, or whole pressure, is to the variation of pressure denoted by 4° as p to d p. Also, the height of the barometer, or whole density, is to the variation of density denoted by 3°, as ę to de. Consequently,

p: 3dp::: 4 dę.

This is nearly the mode of experimenting which I followed. It had been previously practised, through a great range both of temperature and pressure, by MM. Gay-Lussac and Welter; but they do not make the simultaneous variations of density and pressure exactly as 3° to 4°,-which is very nearly the ratio I obtained from a mean of many experiments,—but as 3 to 4.1244. The difference is not considerable, and there is one consideration which, I think, renders it extremely probable that the former is the true ratio *. Sir Isaac Newton has shewn, (Principia, lib. ii., prop. 23,) that if, in an elastic fluid, the cube of the pressure vary as the n+2 power of the density, the particles will repel each other with forces inversely proportional to the nth powers of their distances; and the same may be seen in other elementary works. Now, these experiments

of heat which a body contains. So long, however, as nothing is known, or likely ever to be known, of the absolute heat in bodies, it is useless to have any term for it, and it can only breed confusion to use a term which has commonly a very different meaning.

* I first used a vessel holding about ten gallons of air, and which can be both opened and shut in a small fraction of a second. The opening amounts to 2.88 square inches. I have since repeated the experiments with a vessel of nearly the same size, but having an opening of 25 square inches. With this I intend making a variety of experiments connected with the subject.