9G On Professor Leslie's Formula for determining 



Art. XX. — Demonstration of Professor Leslie's Formula for 

 determining the Decrease of Heat depending on the Altitude, 

 without " a delicate and patient research." * Communi- 

 cated by a Correspondent. 



It is shown in our ordinary treatises on the barometric formu- 

 la, that the height or h— log. B — log. 3= log. -=-, B being the 

 barometric altitude at the lower place of observation, and 3 



that at the higher. Now, for ordinary heights, — cannot dif- 



P 



fer much from unity. Let it be equal to !-}-« then log. 

 (1 -j-n")=M (n 1 — , &c.) If from l + n there be sub- 



» ; * ■ ' 2 3 4 



tracted its reciprocal =1 — n+n- — « 3 -f, &c. there will re- 



r 1-f-M 



1 2 5 



main 1 + n — ; = 2n— ?i-+n :i — , &c. =2 («—— + —, &c) 



' 1 -j- n 2 2 



M 

 Now, if both sides of the equation be multiplied by — ,then 



It is obvious from what has been said, that the first side, 

 when expanded, agrees with the first and second terms exact- 

 ly, and in the third nearly with the common series for the lo- 

 garithm of 1 +«. Hence, when n is a small fraction, the log. 



0+») =-5: 0+"— YZTn) " y ' 



W R 



But h — log. -=-, or log. (1+w); by substituting -=- for 1+n, 



there will result h=— ( — j . . . (A). 



But an elevation of 81 fathoms, by experiment, (Playfair's 

 Outlines, vol. i. p. 295, art. 401,) gives a depression of 1° 

 centigrade. 



Whence if AT denote the variation of temperature — = AT 

 =- — rrf - — r, V Now M for our ordinary atmosphere is 

 * See his Elements of Geometry, p. 459, 4th edition. 



