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density of air suffers a minute and sudden change, such variation of density is to the whole density as three-fourths of the accompanying variation of pressure to the whole pressure; or, p being the density and p the pressure,
dp:p :: dp:p; and
Consequently, one-fourth of the minute variation of pressure must be due to change of temperature; for (art. 1) when the temperature is constant, the pressure varies exactly as the density; and hence because (art. 3) p varies as 448°+t for temperature, we have
5. The equation
3 log. p 4 log. +C;
and if p' and p' be put for the initial values of p C=3 log. p'-4 log. p', and
evidently expresses the relation between the fluxions of the logarithms of the pressure and density of air, when its quantity of heat is constant.
The fluent is
3 log. 2-4 log; or (2)'=()*
and putting t' for the initial temperature, we get
which is the relation between the pressure and density of air, when its quantity of heat is invariable; viz., that the cube of the pressure varies as the fourth power of the density. 6. Also, by integrating the equation
P, we have
for the change of temperature on the common scale solely due to change of density*. Neither this conclusion, nor that
* In the seventh Number of the Quarterly Journal of Science, I adduced a few examples to shew what an incoherent rule Mr. Ivory had given for estimating the change of temperature in air due to a change of
which terminates article 5, necessarily depend on any theory of heat.
7. If the areas which the curves ABC and DEF intercept from the straight line GH, between the same perpendiculars to GH, be everywhere to each other as 4 to 3; and, G being a fixed point, if every three perpendiculars ADH, BEI, CFK, GI GK which make likewise make area BIHA=
FKHD, the curves are hyperbolas.
For if not, with G as a centre, and GH as an asymptote, describe two rectangular hyperbolas, aBc, dEƒ, cutting AH and CK in a, d, c, f. Then all the ordinates of either hyperbola being, as is well known, inversely proportional to their distances from G, the corresponding ordinates of the two hyperbolas are everywhere to each other as BI: EI :: 4:3;
its density, and how directly it is opposed to observation. This I instanced, both so far as regards the inconsistency of the results when we revert the process, and likewise when we compare the change of temperature computed at one operation with what it would be, if computed by the same rule, supposing the whole change of density to take place by successive steps. At present, I only beg to call the reader's particular notice to the circumstance, that the smaller these successive variations of density are assumed, the nearer will the aggregate results or changes of temperature, obtained by the continued application of Mr. Ivory's theorem, approach to perfect consistency; whether among themselves, or when we retrace our steps by reversing the process. Now such continued summation of minute variations of temperature being similar to integration, shews that Mr. Ivory's error consists in his using the fluxion of the time as its fluent ; or at least in assuming the fluent to be proportional to, or a mere multiple of the fluxion. Thus t- being the change of temperature caused by e-g' the change of density, if we take these changes as differentials, Mr. Ivory's theorem, viz.
dt 3 de 448°+t 8
which, with the exception of the constant for, is just the fluxion we had in art. 6; and therefore the integration gives
a self-consistent result, but as different from Mr. Ivory's new law of condensation as light from darkness. The notation in my paper above quoted only differs from this, in that I there used i and unit, in place of t-t' and g' here.
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. GIlog. 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 (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.
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 EIHD (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 HI. 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 BTHA, with its original volume all the while invariable.
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 (arti 1) in the same ratio compounded (art. 3) with the ratio of GI to GK for change of temperature; that is as G H to G K.
Hence, (art. 5) GH ;
But we had also BIHA FKHD; consequently (art. 7) the curves ABC, 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 prèssure 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 assump tion 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.