308 PHILOSOPHICAL TRANSACTIONS. [aNNO 1733. 



whose axes ab, pq, are in any given ratio, as of m to n, have tlie circles apbq, 

 and APBQ, inscribed and circumscribed to it: and if the figure. revolves on the 

 axis PQ, there will be generated an oblate spheroid a/jb^a, with 1 spheres, the 

 greater circumscribed to the spheroid, and touching it in its equator aba, and 

 the lesser inscribed and touching it in the poles p, q ; the solid content of the 

 spheroid being the first of the 2 mean proportionals between the solidity of the 

 exterior sphere, and that of the interior. 



But if the figure revolve on the axis ab, there will be generated a prolate 

 spheroid a/jb^a, inscribed in the exterior sphere at the poles a, b ; and circum- 

 scribing the interior sphere at the equator pqp, its solidity being the second of 

 the above mean proportionals. So that if o and p stand for the solidities of the 

 oblate and prolate spheroids, and s, s. for tlie two spheres; s : o : p : s-r— are in 

 the continued proportion of m : n. And s : 7, or o : s :: m^ : n'. As s : * :: m^ : n^. 



Or we may with Sir Isaac Newton consider the genesis of these solids as 

 follows. 1. Let the sphere apbq be uniformly compressed in the direction of its 

 axis PQ, till that axis is diminished to p^, and the sphere changed into the oblate 

 spheroid. 2. Let this spheroid be equally compressed in the direction of that 

 diameter of its equator, which is perpendicular to pq and ab, or to the plane of 

 the figure; and it will degenerate into the prolate spheroid, whose poles are a 

 and b. 3. Let this last be compressed in the direction of its axis ab, till it is 

 changed into the sphere apbq; and, in each of these compressions, the solid 

 space which the body contains, will be diminished in the ratio of m to 7^. 



Now, as the determination of the earth's figure depends not only on that of 

 the ratio of the centrifugal force, by which a body tends to recede from the axis 

 of rotation, to the power of gravity ; but also on the decrement of gravitation, 

 arising from the body's being in that rotation actually removed to a greater dis- 

 tance from the centre; it is not enough that we know, from the experiments 

 with pendulums, the centrifugal force at the equator to be about ^4-g- of the 

 force of gravity. We need fiirther two distinct propositions; one to determine 

 the attractive force of a spheroid at its pole ; and the other to determine its at- 

 traction at the equator. The first of these we have in Princip. lib. J, prop, gi, 

 and the second has been supplied by several authors. But Sir Isaac, who seldom 

 does any thing in vain, found that he could, by one of his artifices, make that 

 gist proposition serve likewise to determine the attraction at the equator, by the 

 following argument. 



Let G be the attraction of the exterior sphere at a; and let the decrement of 

 that attraction, when the sphere is diminished into the oblate spheroid Apsq, be 

 d; and S the decrement of this last attraction, when the oblate spheroid is dimi- 

 nished into the prolate, whose poles are ab; then is d nearly equal to S; the 

 difference of the axes of the generating ellipse being small. For the attractive 



