AT THE SURFACE OF THE EARTH. 685 



very close to the surface. Let E be an element of this surface lying around the radius vector, an 



element which for clear ideas we may suppose to be a small circle of radius s, and let s be at the 



same time infinitely small compared with a, and infinitely great compared with the distance between 



dV' d V" 



the points. Then the limiting values of — — and — — will differ by the attraction of the element 



dr dr ■' 



E, an attraction which, as follows from what was observed in Art. 13, will be ultimately the same 

 as that of an infinite plane of the same density, or Stp^*. The mean of the values of and 



dr dr 



will express the attraction of the general coating in the direction of the radius vector the 

 general coating being understood to mean the whole coating, with the exception of a superficial 

 element lying adjacent to the points where the attraction is considered. Denoting this mean by 

 dV^ 



— — , we get, on putting r = a. 



= -2,r2- 



dr 2i + 1 



This equation becomes by virtue of either of the equations (28) 



y^ = -I% (30) 



dr 2a ' 



which is a known equation. Let either member of this equation be denoted by — g''. Then 

 gravity will be increased by g", in consequence of the attraction of the general coating. 



20. But besides its direct effect, the attraction of the coating will produce an indirect effect by 

 altering the sea-level. Since the potential at any place is increased by V^ in consequence of the 

 coating, in passing from what would be a surface of equilibrium if the coating were removed, to the 

 actual surface of equilibrium corresponding to the same parameter, jthat is, the same value of the 

 constant c in equation (1),} we must ascend till the labouring force expended in raising a unit of 



mass is equal to V^, that is, we must ascend through a space — , or — nearly. In consequence of 



^ . 

 this ascent, gravity will be diminished by the quantity corresponding to the height G"' F^, or li 



suppose. If we take account only of the alteration of the distance from the centre of the earth, 



2h' 2V^ „ 



this diminution will be equal to G. — , or , or ig , and therefore the combined direct and 



a a 



indirect effects of the general coating will be to diminish gravity by 3g". 



But the attraction of that portion of the stratum whose thickness is h', which lies immediately 

 about tlie station considered, will be a quantity which involves /*' as a factor, and to include this 

 attraction we must correct for the change of distance // by Dr. Young^s rule, instead of correcting 

 merely according to the square of the distance. In this way we shall get for the diminution of 



gravity due to the general coating, not 3g", but only 4 I I ] g" - g'l '"' '^g' suppose. If 



a ■■ p.: S: II, we have /c = r . (H nearly. 



* This roult readily follows from equations (2R), which give, 



on putting r=at — . , - = 47r 2 ^( = 4Tr3. ThiR difference of 



dr dr 



attraction at points infinitely close can evidently only arise from 



the altracliori of the Interposed element of surface, which, being 

 ullimutely plane, will net equally at both piiintu \ and, thcrefiire, 

 the altractiim will lie in each case 'Jiri, and will act outwardj in 

 the iitst case, and inwards in the second. 



Vol. VIII. rAiiT V. 4T 



