212 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738 



Now if we have regard to the smallness of the line nv, and observe how 

 little angle vnc will differ from a right one, we may perceive that the diameter 

 CN contains the same angle with the perpendicular nx in n, as the diameter 

 CN with the perpendicular at v; that is, that the angle ncv is the same as the 

 angle cnx; so that instead of n we may take — . Therefore the foregoing ex- 

 pression of the attraction of the ellipsoid BEbe, acting according to the direc- 

 tion ex, on a corpuscle placed in n, will be -f * x — + — f^ X — . 



' ' ' 5+p CN 5+q CN 



Problem VII. To find the Direction of the Attraction of a Corpuscle n 

 towards the Ellipsoid. — 12. By the second Problem we shall find the attraction 



of the spheroid according to on to be —^ \- ^^ — , by expunging what 



may be here expunged. Then by taking a 4th proportional to these 3 quanti- 

 ties, the first of which is the attraction according to cn, the 2d is that accord- 

 ing to ex, and the third is the right line cn; there will arise 



5 + p "*" 5 + q _ 

 rl+P , +o X CX CL. 



3+p "^ 3 + 9 



Whence we shall have ni for the direction required, of the attraction of the 

 corpuscle n. 



1 3. If we suppose p z= q ■= o, that is, if the spheroid be homogeneous, we 

 shall have ci = ^cx ; which agrees with what Mr. Stirling has found, in that 

 curious dissertation he has published in the Philos. Trans. N° 438. 



Part II. The Use of the foregoing Problems, in finding the Figure of 

 Spheroids, which revolve about an Axis. — 14. Let us now suppose, that the 

 foregoing spheroid bneZ^c, (fig. 13) which is still composed of strata of differ- 

 ent densities, revolves about its axis Bi, and that it is now arrived at its per- 

 manent state. It is plain that the particles of the fluid, which are on its sur- 

 face, must gravitate according to a direction perpendicular to the curvature 

 bne; for without this condition there could be no equilibrium. 



We shall now inquire, whether the elliptic figure we have ascribed to our 

 spheroids can have this property, and to produce this effect, what must be the 

 relation between the time of revolution of the spheroid and the difference of 

 its axes. 



Let us then put (p for the centrifugal force at the equator, and the centrifugal 



'111 ^XPN ^X CX 



force at n will be , or , because 2pn X a, = ex. 



CE ' 2CE X a. 



