MEASUREMENT OF HEIGHTS. 521 



vation. In order to correspond approximately to this view, and 

 at the same time to give the integral the most simple form pos- 

 sible, Laplace assumes 



{I + k tf + -^4^ 



^ ' a + X 



to be constant for all corresponding values of t and x, and deter- 

 mines the constant i, so that it may satisfy the two observed 

 temperatures t and t'. Hence 



(1 + ktf -{- iX = (1 +kTY + iH = (1 + kr'f + ill' 



where I have written X, H and H' for , — ; — j-, j-i 



a + X a + n a + H. 



We obtain thereby 



* = H'^n ' 



and 



^x / « ^"- ' 2^ 



and further, 



\a + x/ 



d X 2^ ,^ 



whence the integral taken from h to h' is 



2k , ,, H' - H 



— (t - t'^ - 



We have thus, in accordance with Laplace's assumption of the 

 law of the change of temperature, transformed the foi'mula (5.) 

 into 



, F 1 H'-H 



log p = y. -—-,-, . . . . (6.) 



1 + k-^- 



3. 



1 have hitherto considered the air as dnj, and have still to 

 take into account the aqueous vapour which it always contains. 

 If, in a circumscribed space, the mixture of the dry constituents 

 of the atmosphere exert on the circumscribing surface the pres- 

 ure p, the aqueous vapour the pressure 2^,> a"d if the specific 

 gravities of the two be respectively denoted by D and d^ D, and 

 of the moist air which results from their mixture by D', then 

 according to equation (1.), 



