﻿354 Mr. M. Hall on the Determination of 



The value of a is to be found from the equation p =ag p , 

 where p is measured by means of the mercurial barometer. 

 Since k is constant, a will vary with the latitude; but its mean 

 value for the whole earth is about 4"94 miles. 



Thus the whole pressure of the atmosphere is equal to the 

 whole pressure of a homogeneous atmosphere whose density is 

 p and height a ; and in consequence a is termed the height of 

 "the homogeneous atmosphere." 



Let r = 3958*4 miles, then a sphere of radius r will be equal 



47r 

 in volume to the earth, and therefore -^- Ha + r ) 3 -- rjjj- will be 



the volume of the homogeneous atmosphere; and its weight 

 will be s 7r 9oPo\{ a ~^ r o) 3 ~ r l\ ' which must be carefully distin- 

 guished from its whole pressure on the earth's surface, which is 

 evidently equal to 4i7ra(/ p r*. 



The weight 3 7rg Po{( a ~t~ r o) 3 ~ r l\ is approximately equal to 



1 -! > ; and if h be the height of the exterior sur- 



face of the atmosphere above the surface of the earth, we get 



C a 1 C ro+h 

 ^ a ffoPo r l\_ l +—j = ] 4firgpr*dr, . . (6) 



an equation to determine h. 



But ^r 2 =^ rj, and p = p €~a\ V ) from (5), and r = z + r ; 

 therefore equation (6) becomes 



I r oJ Jo 

 Again, 



€ aV. r Q J =6 a t ear Q z=e « 1H I 



V ar J 

 approximately ; we can now integrate ; and dividing by a, 



1 + - =1 — e~« + — |2a 2 -e-a(A 2 + 2^ + 2a 2 )}; 



therefore 



a 



r o 



«< 1+ >; 



L ar n J 



and taking logarithms, we get, finally, 



g-log/" 1 h *+ 2ah +* a \ ...... (7) 



ar f 







which may easily be solved by approximation; the resulting 

 value of h is 33'4 miles. 



