274 ' BIB.ECT ATTRACTION OF A SPHEROID. 



1 i.V" X 

 x^ (2 — 2x^] is less tlian , that Is, — ' 



4 x^ X 



. — of which the fluent is found as before U x* 



— y»j. x'^ — _?^^ ^ (l — .g ^-^ . and this becomes -rV while x 

 increases from to 1, being to 2, the attraction of the 

 whole shell, as yV to 1 ; but if the radius of the sphere be 

 1, and the ellipticity e, the attraction- of the shell will be' 



to that of the sphere as — to 1, ?? being the naean- den- 

 sity of the sphere, compared with that of the superficial 

 I parts, and the attraction of the spheroidal prominence will 



4 ^ 

 be expressed by — , tlmt of the spliere beins; unitv. 



Polar & equa» The depression below the circumscribed sphere is equal, 

 ona a. rac- ^^ ^-^^ meridian, to the elevation above the inscribed sphere: 

 but vanishes at the equator, being every where propor- 

 tional to the square of the sine of the lati-tude ; so that the 

 mean depression of each of an infinite number of rings, of 

 which any point of the equatm- is the pole, must be half as 

 great as the elevation of the corresponding rings parallel to 

 the equator; and- the w^hole deficiency is equal to half @f 



'2 e 

 the whole excess, that is, to — : consequently, the re- 



13 e 

 maining attraction of the shell is — — , from which we 

 ^ 5n 



must deduct the diminution of the attraction of the in- 



J3 e 



scribed sphere 2 €, and the whole will become 1 4 



4 <' (ye 



2 (?, which subtracted from 1-1 leaves 2e — ' for 



5 n 5 n 



the excess of the immed-iate attraction at the* pole above 

 tlie equatorial attraction; to v.-hieh if we add' the ceiitrifu-- 

 ga.1 force/, the whole diminution of" gravity g- will be 2 e — 



'-— .+ /; but since e v/as before found to /"as 1 to 2 — — - 



, on . , 10 ?i — 10 71 — 9 

 or z= -r Tv . /, we have . e jz--- —-- . /, 



and 



