VOL. XL.] PHILOSOFHICAL TRANSACTIONS. 213 



By resolving this centrifugal force according to the perpendicular to cn, we 

 shall have ^ ^ ^^ ; to which adding -^^-^-; X — + ~-, X — , found by 



2<» X CE ° 5 + p CN ' 5 + J CN •' 



by prob. 5, will give the whole force of the body n, according to the direction 

 ex, when the spheroid is turned about its axis. But because this body, by 

 virtue of the attraction according to cn, and the force according to ex, ought 

 to have a perpendicular tendency to the superficies; we shall have this analogy, 



CN : ex :: -f- — + -I- : — X \- -^, X §- X — . And 



3+P 3 + (j 2» ce' 5 + p cn' 5 + q cn 



hence, because cn and ce may be assumed as the same on this occasion, it 



Will be ip = — == + 



3 + p X 5 + p 3 + q X 5 + q 



The Spheroid being supposed ellipical. Bodies will gravitate perpendicularly 



to its Surface. 



And as in this value of the centrifugal force, no quantity enters but what 

 will agree to any point n; we may therefore conclude, that when our supposed 

 elliptical spheroid performs its rotation in a proper time, so that the centrifugal 

 force at the equator may be as before ; then the centrifugal force in any other 

 place n will be such as it ought to be, to cause bodies to gravitate in a direction 

 perpendicular to the surface. 



The Expression for the Gravity at any Place on the Spheroid. 



15. If we now consider, that ed (fig. 14) being taken for the centrifugal 

 force in e, then will mn express the centrifugal force in n, and consequently 

 MI will be such a part of this force as acts according to nc ; we shall have 



+ = — == to be subtracted from the attraction at n. Hence 



3+px5+p 3+qx5+q 



•Zcfe'-^f 2p- Wcfi^e'+f Scfue' + f 2cge' + ^ 2y - lOcg^e'+f 



3+p 3+~p X sTp 3+px 5 + p 3 + q "*" 3~+Tx 5TT 



' scgxe' "'"^ 



— ■ - Will be the gravity at n. ~ 



S + q X 5 + q ^ ^ 



The Gravity at the Equator. 



16. In this value making x = «, we shall have 



2c/e'+>* 2F^';/-«e'+^ , 2cg-e' +? , 27-^c?«e'+^ , 



-r—. — + ' ~^^= + + — =— for the gravity at the 



3+p ' 3+px S+p ' 3 + q ' 3 + qx 5 + q ^ ■> 



equator. 



17. If we subtract the value of the gravity in n from the value of the at 

 traction or gravity at the pole, in art. 3, we shall have 



