THE BATHOMETER. 125 



thin slices by planes perpendicular to that line. Each slice 

 is composed of concentric rings of the sectional area 



dh . z . d a 



dh . dx = 



cos a 



and of the capacity 



2 TT . z . sin a . dh- . z . d a 



> 



cos a 

 and the differential of the attraction is 



2 TT . z . sin a . dh . z . d a . cos a 



d . d . A' = 



=. 2 TT . d X . sin a . d a, 

 which, integrated between the limits h and o and a and o, is 



as the total attractive force exercised by the uppermost 

 portion of the earth to the depth h. 



For small values of h, 



A' 2 TT . h (2). 



Substituting 211 for A in (1), there is obtained 



A = 1 R TT (3). 



The proportion between the upper segment and the 

 whole earth is 



-. 



If sea-water were without weight, the attraction at P 



would be diminished in the above ratio of depth of water 



2 

 of ^ R, but the real influence of the depth, taking the 



o 



specific gravity of water into account, is very nearly as the 

 depth to the earth's radius. 



If this calculation is correct, it follows that the total 

 gravitation of the earth will diminish in passing over the 

 surface of the ocean in the exact proportion of the depth 

 of the sea to the radius of the earth, and as the radius 



