Mr. Ivory on the Elliptic} ty of the Earth 

 fj,d. a * J_ fed (a' A?) _ vie 5 v _ r , i 



~ a» 6 ' a' - 175 * I JZ aC \ 



+ + . iu-f S d(- A -^) \ ■ - 4 . fo-° 3 



T 3 ( J * V a 1 J S 35 «3 



I8e fed(a*e ) _3_ fjd{££) _4_ 



175 ' a 5 35 ' a 5 + 35 ? 



All the integrals begin at the centre of the fluid mass and 

 terminate at the surface; F, G, H, denote the whole inte- 

 grals accumulated between the limits mentioned ; and q = 



— - — x ~- > T representing the time of a revolution, and 



7T = 3-1416. 



If we reject all the terms containing <?-, there will remain 

 only the first line of the first equation, which is no other than 

 the solution obtained by Clairaut. If we make § constant, we 

 shall find A = ; and the first equation will give the relation 

 between e and q, pushed to quantities of the second order on 

 the supposition that the fluid is homogeneous. The quantity 

 A, therefore, begins at the surface of the nucleus and extends 

 to the outer surface of the fluid. From what has been said 

 there is no doubt that the two equations are consistent and 

 possible. 



But if the equations be possible, they are not easily solved ; 

 and besides, we are ignorant of the law of density in the in- 

 terior of the earth. All therefore that can be done, is to 

 form the expressions of the radius of the earth at any latitude, 

 and of the force of gravity which prevails there, in order to 

 compare them with the measurements of the meridian and the 

 variation of gravity as determined by means of the pendulum. 

 Now the polar semiaxis being unit, and X the latitude, if we put, 



the radius of the earth, or r, will be thus expressed, 



r= 1 + ecosx + (|-r - -f§As 2 ) sin 2 2 A: 



and if g and G be the gravity at the latitude K and at the 

 equator, then <p denoting the proportion of the centrifugal 



force to gravity at the equator, and — <p — "00865 ; we shall 



have, 



j = G : jl'+(4f-.+-f-^f i «+ |-A £ «)sin*A 



it 



