246 Mr. J. Ivory o/i the Consliiuiion of the Atmosphere. 



necessary to remark, that in many of the instances the differ- 

 ence of temperature is inconsiderable, and that a variation of 

 a degree or two at either extremity of the height measured 

 will materially change the divisor, and occasion a great un- 

 certainty in the quotient. 



If we expand the logarithms in the divisor of the formula 

 (b), and reject the powers above the square, we shall get 



log. , ; , = a(T— t') X \l ^ \ = ^ -TTT • 



»!-!-«•/ ^ ' i 2 5 T -\-t' 



Wherefore, by substitution, 



But, from the relation that subsists between the difference of 

 temperature and the height in the class of atmospheres we are 

 considering, we have ul x 5 x {t — t') = x, and 



= 5. 



al (^T — t') 



Hence, by equating the equal quantities, we obtain 



X = ^ X log. y X j 1 + «. ^^^ I , (c) 



which is no other than the usual formula for finding any 

 height in the atmosphere by means of the barometer. 



It has now been proved that air of a given elasticity, or sub- 

 jected to a given pressure, has the same temperatui'e in the at- 

 mosphere vve are considering as in that of nature ; and the last 

 formula proves that the same pressure likewise takes place in 

 both cases at the same altitude. It appears therefore that the 

 equations (a) represent the real state of the earth's atmosj)here 

 as nearly as the experimental knowledge in our possession 

 enables us to judge. 



It deserves to be mentioned, that in the investigation of the 

 barometrical formula (c), the particular number 5 disappears, 

 and the result is therefore true for all values of the general 

 symbol 7ii. That formula is a general property of all the at- 

 mospheres in which the loss of temperature is proportional to 

 the height, whatever be the rate of the gradation of heat. But 

 it likewise belongs to many other atmospheres, besides the 

 class mentioned ; and we may affirm generally, that the usual 

 barometrical formula is an approximation independent of any 

 particular law of the gradation of heat, and that it is true in 

 every atmosphere in which the known laws of elastic fluids are 

 supposed to prevail. If we consider that in practice t and t' 

 are found by observation, and that these quantities affect the 

 height measured only in a small degree, it will be admitted 



that 



