138 Count Paul de Saint- Robert on a Barometrical Formula 

 the tirst two terms, we have the formula 



^i og ^_ J( , o _ r, + ,_ 2 v- 



x — 





(4) 



for calculating the difference of level x between two stations 

 where the barometric pression and the temperature are known. 

 As the term between crotchets is very small, it is convenient 



to introduce the development of log -y, which is 



l0g t~ t + t + 3 (/ + 3 ' 



a very convergent series. 



The barometrical formula will then become, neglecting the 



second power of the fraction -° — - as very small, 



t +t 



The first term of the value of x is due to a uniform decrease 

 of 'temperature with the altitude, and the second is due to the 

 retarding ratio of that decrease. 



We must observe that if we expand log-j in the first term of 



x, we obtain 



X ~ g\ 2 ) {0 °p ff \ 2 r%p 3\t + t) 3° t + t' 

 neglecting only the third power of -^ — -, because the quantity 



by which it is multiplied is much larger than the coefficient b. 



The first term is the well-known formula given by Laplace. 

 Therefore if we denote by X the height furnished by Laplace's 

 formula, the correction to be applied to it, to take into account 

 the retarding ratio of decrement of heat, is expressed by 



3 \t +tJ 3° t Q + t 



This correction is very small for small differences of the tem- 

 peratures of the two stations— that is to say, for small differences 

 of level — but becomes of consequence for considerable differences 

 of temperature, and therefore of level. : 



