18 On the Height of Mountains, Headlands, fyc. 



Dividing the latter series by d n , we have 



Altitudes, 0, 1, 2, 3, 4, 5, &c. 



Corresponding densities, 1, d~ l , d~ 2 , d~ 3 , e? -4 , d~ 5 , &c. 



This is strictly analogous to the property of logarithms. In 

 fact, the several altitudes form a peculiar system of logarithms, 

 of which the reciprocals of the corresponding densities are the 

 natural numbers ; from this circumstance they have been denom- 

 inated atmospheric logarithms. From a similar circumstance, the 

 Napierian are termed hyperbolic logarithms, because they express 

 the areas contained between the asymptote and curve of an hy- 

 perbola. We shall write these atmospheric logarithms with large 

 letters — thus, "Log." — to distinguish them from the Briggean or 

 common logarithms, which are written "log. , " or simply "log." 

 and also from the hyperbolic, which are denoted by " n log." 



Let a, A represent any two altitudes, and d, D their corres- 

 ponding densities. 

 Then will A= - Log. D, 

 and a=— Log. d ; 



d 

 .'.A — a=LoG. d — Log. D=Log. ^y 



Now it is a well known property in logarithms, that by assum- 

 ing different values for the base, there will be as many different 

 systems of logarithms ; and it is equally well known, that in all 

 the various systems of logarithms, the logarithms of the same 

 numbers can be converted from one system to another, by a con- 

 stant multiplier or modulus. 



The object of our present enquiry is to determine a constant 

 multiplier that shall convert the common logarithm of a number 

 into the atmospheric logarithm of the same number. To accom- 

 plish this, let 



d d 



Log. y.=:rlog. ^y 



'. A— a — x\o% 



d 

 D 



Then making a = 0, or which is the same, if we suppose d to rep- 

 resent the density of the atmosphere at the surface of the earth, 

 we shall have 



d 

 A=.rlog. jj- 



In order to find x, let us take the height of a homogeneous at- 

 mosphere, when the temperature shown by the thermometer is 



