Formula for Mountain Heights. 441 



Comparing equation (3) with corresponding equation of Poisson's "Mecanique" 

 (vol. II. chap. V. article 542, Paris edition, 1811), viz. 



dp _ — gr"^ dz 



p a (1 + ax) ' (r + zy 



and recollecting that, in Poisson's equation, the product ax has the same 

 signification as 1{T — 32) of our equation, x being the number of centigrade 

 degrees above freezing point, and a being the expansion of gas for each 

 centigrade degree : a being, in Poisson's equation, as in ours, the modulus 

 of elasticity of dry air (the only air considered by Poisson) ; also p having the 

 same signification as tt, and the other characters of Poisson's equation being the 

 same as in ours. It appears that, in Poisson's equation, the denominator p of 

 the left-hand side of the equation is too great; by way of compensation, 

 Poisson, following the recommendation of La Place, increases the denomi- 

 nator of the right-hand side of the equation, by increasing the quantity a, — 

 making it 0'004, instead of 000375, its real value. Now, though this increase 

 of the value of a may improve Poisson's equation, when the temperature is 

 above the freezing point, it unquestionably increases the error when the tem- 

 perature is below the freezing point, because, under such circumstances, x 

 in Poisson's equation becomes subtractive, and the denominator of the rightr 

 hand side of Poisson's equation, instead of being increased, becomes diminished. 

 Moreover, the substitution (even when the temperature is above the freezing 

 point) of 0'004 instead of 0'00375 is, at best, but a guess as to quantity, and, 

 like all guesses, more or less uncertain: hence the necessity of some effort to 

 improve the formulse given by La Place and Poisson. 



4. Let us now proceed to integration. In order to do so, we ought to be 

 acquainted with the law of variation of temperature of the atmosphere between 

 the stations of barometric observations ; also the law of variation of the elastic 

 forces or tensions of vapour of water between the said stations. In the absence of 

 correct information relative to the first law, it is assumed that the temperature 

 of the atmosphere between the stations varies uniformly; a mere supposition 

 (sometimes true, sometimes far from it). According to such assumption, we have 

 to replace the variable {T — 32), by the arithmetic mean of the temperatures 

 of the stations as ascertained by the detached thermometers, that is, to replace 



