ON THE MATHEMATICAI. THEORY OF FLUIDS. 229 



pressure of the atmosphere at any assigned altitude above the 

 earth's surface, in other words, to solve the problem of the ba- 

 rometric measurement of heights, a second equation, expressing 

 the relation of the temperature to the density or the pressure, 

 would be required. For want of such an equation, Laplace as- 

 sumes, in investigating his formula for the determination of 

 heights by the barometer*, that the temperature is uniform, and 

 equal to the mean of the observed temperatures at the higher 

 and lower stations. This supposition^ as Mr. Ivory has shown f, 

 conducts to the same barometric formula as would result from 

 supposing the decrements of atmospheric temperature to be 

 equal for equal increments of height above the eai-th's surface. 



The only attempt I know of which has been made to collect 

 the law of vai-iation of temperature at different heights in the 

 atmosphere from observations, is that by Mr. Atkinson in his 

 Memoir on Astronomical Refractions contained in the second 

 volume of the Transactions of the Royal Astronomical Society. 

 His object is to arrive at the law by a consideration of as many 

 recorded observations as could be procured of temperatures in 

 different latitudes and different elevations (principally those of 

 General Roy J and Baron Humboldt §), by a discussion of which 

 he comes to the conclusion, that for equal decrements of tem- 

 perature the increments of height are in arithmetic progression. 

 The following is the table of results given at p. 189 of the Me- 

 moir, from Humboldt's Observations in South America : 



Height. Depression of Therm. 



3724 Feet 14°-070 Fahr. 



6740 23°-310 



9029 30°-070 



10790 34°-7l5 



15744 49°-620 



19286 57°-380 



The same arithmetic progression results from the observations 

 in Europe as from those in South America, and the general em- 

 pirical formula connecting the height h (expressed in feet) and 

 the depression n (expressed in degrees of Fahrenheit) below the 

 temperature at the earth's surface, is the following : 



h = ^251-3 + I (« - 1)}«. 



'este, liv. x. chap. iv. §. 14. 

 Transactions, 1823, p. 455. 

 '77, part ii. p. 653. 

 § Memoir on Isothermal hmes, in the Memoires d'ArcueU, iom. in p 46'>. 

 translated m Edin. Phil. Journ., vols, iii., iv., v. ^ ' ' 



* Mecanique Celeste, liv. x. chap. iv. §. 14. 

 t Philosophical Transactions, 1823, p. 455. 

 X Phil. Trans. 1777, part ii. p. 653. 



