VOL. XL.] FHILOSOPHICAL TRANSACTIONS. 123 



by a very small quantity, which for greater perspicuity I shall call infinitely 

 small. If tins spheroid be conceived to be of a fluid and homogeneous matter, 

 and revolved about the axis ha in correspondent time, that the gravity of the 

 column CE may be equal to the gravity of the column ac, that is, by the New- 

 tonian principles, the attraction in e, neglecting the centrifugal force, may be 

 to the attraction at a, as ca to ce : I say, that all the columns on, wanting an 

 infinitely small quantity of the second degree, will preserve an equilibrium with 

 those two columns; that is, the attraction on n, neglecting the centrifugal 

 force, simply in the direction on, is to the attraction on a, as ca to on. 



For the demonstration, the same notation will serve as in the preceding pro- 

 position: first find the centrifugal force at e, which may agree with the equili- 

 brium of the columns ce, ca. Then say, as \pa -)- -^pam — f: fpa -\- -J^pam 

 :: 1 : 1 -|- »n, hence is found/= -^pam. 



Then for exhibiting the gravity at n composed of the attraction, omitting 

 the centrifugal force, find the centrifugal force at n, or, which is the same, 

 on M upon the sphere, which must diff^er from each other only by an infinitely 

 small quantity of the second order, if de be supposed to express the centrifugal 

 force y at e, then mn will express the centrifugal force at n, for the centrifugal 

 forces are as the radii, when the times of revolution are the same, and by the 

 property of the ellipses it is de : nm :: cb : MP. 



But if the centrifugal force act in the direction np, it must be reduced to nc, 

 and NO will be the remaining part. Therefore the centrifugal force at n, or at 

 M, is to the centrifugal force at e, or at d, as no is to de. Therefore the ex- 

 pression for the centrifugal force at n will he'-Jt^pan, and consequently the ex- 

 pression for the gravity there will be ^pa — -^pan -{• -^pam — -^pan, or \pa 

 — ^pan -\- -^pam. 



Now to find the centrifugal force at n, which results from the equilibrium of 

 the columns, the gravity at a must be to the gravity at n, as nc to ac; but the 

 gravity at a is ^pa -f- -^pam, which expression being drawn into or J — n, 



after reduction it becomes ^pa — ^pn -|- -^pam, and is the same expression as 

 that above. 



Hence we see that there can be only an infinitely small difference between 

 the figure which the earth ought to have by the Newtonian hypothesis, and 

 the ellipsoid. For as the quantity de is about the 230th part of ac, in the 

 preceding computation we neglect only a quantity of the same order with 





