in geometrical Progression for equal Increments of Heat. 245 

 B' would still produce the former absurdity. Hence ABC 

 can only be a hyperbola. T^Tt^r' 



Othervoise direcfli/.—'Lei a new ordinate cut the area iJlKU 

 in the given ratio, and consequently the differences m the lo- 

 garithms of the three abscissae concerned are ni that ratio. 

 Do the same with area AHIB, and the like follows. In short, 

 it is evident that in the area AHKC, new ordinates might be 

 interpolated successively, each to divide an interval between 

 two ordinates in the foresaid ratio, till the whole space be- 

 tween AH and CK, no matter how extensive, be divided 

 into parts less than any given area, and each of them, orits 

 numerical value, having the same ratio to the corresponding 

 difference in the logarithms of the abscissa, that the whole 

 space AHKC has to log GK-log GH. 



Hence the variations in the area of the curve ABC are 

 everywhere proportional to the corresponding variations in 

 the logarithms of the abscissae; and, therefore, so are any 

 spaces into which that area may be divided by ordinates: 

 which is a property characteristic of a hyperbola having G for 

 its centre, and GH for an asymptote. 



I shall now distinctly state the physical principles employed 

 in this investigation, and which are essentially the same as 

 those used by Laplace and Poisson. 



1. The law of Boyle and Mariotte, that under the same 

 temperature the elasticity or pressure of air is as its density, 

 or inversely as its volume ; and consequently while air under- 

 goes the same change of temperature, its volume varies, under 

 a constant pressure, precisely in the same proportion as the 

 pressure would do, were the air confined in an inextensible 

 vessel. 



2. It has been ascertained by Mr. Dalton, M. Gay-Lussac, 

 &c. that on heating air, under a constant pressure, from 32° 

 Fahrenheit to 212°, its bulk acquires an increase of three- 

 eighdis. Such increase in the common graduation of the air- 

 thermometer is divided into 180 equal parts or degrees for 

 Fahrenheit's scale; and the like divisions, corresponding to 

 equal variations of bulk, are continued both above 212° and 

 below 32°. Now as three-eighths are to 180°, so is the whole 

 bulk at the freezing point to 480° ; and therefore there can- 

 not be more than 480 degrees below 32° on the Fahrenheit 

 scale of an air-thermometer. 



3. The freezing-point being marked 32°, it is obvious that 

 4S0 degrees below this will be at - 448°. The bulk of a given 

 mass of air, therefore, under a constant pressure, varies as its 

 temperature reckoned from -448 on the common scale; that 

 is as 448° -f/, the degrees of Fahrenheit being /. Hence by 



