and furnishing a true analytical representation of the phenomena. This was done by M. Bettancourt, and the curve constructed from his equation has a point of inflexion at about the 102° of Reaumur, as it ought to have, because the second differences of the barometrical measures of the elastic force became negative at that temperature.

60. In a similar manner M. Bettancourt made experiments on the strength of the vapour from alcohol or spirit of wine; constructing the curve and deducing the requisite analytical formula. This curve had likewise a point of inflexion at about 88° of Reaumur, the second differences in the table of barometrical measures becoming then negative. From a comparison of the experiments on the vapour of water with those on the vapour of alcohol, a remarkable conclusion was derived: for it appeared that, after the first 20° of Reaumur, the strength of the vapour of spirit of wine was to that of the vapour of water, nearly in the same constant ratio of 23 to 10, or 7 to 3, for any one and the same degree of heat. Thus, at the temperature of 40° of Reaumur, the strength of the steam of water is measured by 2.9711 Paris inches in the barometer, and that of vapour of alcohol by 6-9770, the latter being about 24 times the former.

61. The equations to the curve of temperature and pressure, denoting the relation between the abscissa and ordinates, or between the temperature and the elasticity of the vapour, as given by M. Bettancourt, were of the following form.

Where y represents the height of the column of mercury which measures the expansive force, r the corresponding degrees of Reaumur's thermometer, and the other letters certain values which are assigned to them in the investigation.

62. But M. Prony, in the 2d volume of his Architecture Hydraulique, has thrown these equations into a rather more convenient form, though analogous to those of Betancourt. His formula for the vapour of water is this,

The method which he followed consisted in satisfying the results between 0° and 80°, by means of the two first terms, and interpolating by means of the other two, the differences between the observed values, and those computed by the two first terms, from 80° up to 110°. In this manner he succeeded to express so exactly the observations in their whole extent, that

the curves of the calculus and the experiments were only distinguishable the one from the other by such little anomalies, as were manifestly the effect of some trifling though inevitable errors in the observations, and in the graduations of the scales in the apparatus. He afterwards employed an equation of three terms, giving to the different coefficients the following values:

log. p, 0.0692259 log.00202661 log.g=0.0120736 78601007

.log.

log.

9369271

log. 9369248

it satisfies not only the numbers employed in its formation, but all the intermediate observations, as may be concluded from the following table, which exhibits to every 10 degrees of Reaumur's thermometer the barometrical results both of obvervation and the calculus.

The anomalies are generally much more minute than in the formulæ of four terms: we may therefore regard the equation just preceding the table, which is more simple than that of Bettancourt, as representing the phenomena and measuring the effects of the expansive force of the steam of water with all desirable accuracy. M. Prony remarks, that the smallness of the coefficient, will allow the term to be neglected in reckoning between 0 and 80°; and thus from the temperature of ice

up to that of boiling water, the equation of two terms alone

will suffice, that is to say....y +1.

63. M. Prony's equation for the vapour of alcohol comprises 5 terms originally but in most cases three of those terms will give results sufficiently accurate. The numeral values of the coefficients are as below:

These numbers cause the experiments and calculus to coincide very nearly, when introduced into the equation.

y = μ, s, " + ",, 8, + !,,, .,, +14

The magnitude of the anomalies will be seen by inspecting the following table.

Thus the formula for the vapour of spirit of wine is found as simple as that for the vapour of water, without ceasing to represent the experiments with all desirable exactness. But more than this, we may retrench one of the variable terms; for in the first degree has no greater value than 18, and when ris 2, 3, or any other positive value, this third term may be af ely neglected. The equation therefore is reduced to

a form much more simple than Bettancourt's original equation, and indeed more simple than Prony's improved equation for the vapour of water.

64. To save the trouble of investigating the strength of the vapour by these formula for every separate case that may occur, we add a table (calculated from these principles) in which the strength of the vapour both of water and of spirit of wine is shewn for every degree of Reaumur's thermometer up to 110°, or for every 24 degrees of Fahrenheit, from 32 to 280: the strengths are expressed, not in English or in French inches upon the barometer, but in terms whose unit is the medium pressure of the atmosphere, supposing that medium equivalent to 29.9 English, or 28 French inches of mercury. The pressure upon a square inch in pounds averdupois corresponding to any temperature may be found by multiplying the corresponding number taken from the table by 14.75: and the pressure for any intermediate degree of Fahrenheit may be found pretty nearly, by proportioning, as is usual in tables of Logarithms, &c.