VOL. XXXVII.] I'HILOSOPHICAL TRANSACTIONS. 523 



plane. And since, in this hypothesis, the centrifngal force can have to gravity 

 all ratios, from nothing up to the ratio of equality, it appears that the equato- 

 rial diameter can have to the axis of rotation all ratios ; and that the spheroid, 

 which in this hypothesis is always an ellipsoid, can be all ellipsoids, from a per- 

 fect sphere down to the circle. But in this hypothesis also the centrifugal force 

 cannot increase further. 



If gravity be taken reciprocally proportional to the square of the distance, 

 then will n = — 2, and CA : cp :: 2Jj +/: 2p. From which it appears that, 

 in this hypothesis, the centrifugal force may continually increase, or, which 

 comes to the same thing, that the rotation can be continually quicker, without 

 the parts of the spheroid being dissipated. 



Scholium. — But, of all these hypotheses, there cannot be usurped any one 

 as really given in nature ; as I do not suppose that the interior parts of bodies 

 gravitate towards any one centre, according to any proportion of the distances 

 from that centre. The attraction of the parts depends on the form of the body, 

 and interchangably the form depends on the attraction. Therefore all these 

 determinations are more mathematical than physical. Hence it is that Newton, 

 in the determination of the axis and equatorial diameter of the earth, found 

 the ratio different from that of Huygens and ours, namely that of 22g to 

 230. That great man neglected the solution merely geometrical from hypo- 

 thesis, that he might give it more agreeable to nature. 



Problem II. — Supposing that the matter flowing about an axis taken with- 

 out the fluid, be attracted towards a centre posited in that axis, by a force 

 proportional to any power of the distance from the centre ; while at the same 

 time, because of the mutual attraction of the parts of the fluid, there is made 

 another attraction towards another centre taken within the fluid, which in any 

 section made through the exterior centre perpendicularly to the fluid of rota- 

 tion, is proportional to any power of the distance from the interior centre : to 

 find the figure the fluid will take. 



Let AVPadotA, fig. 7, be a section of the fluid gyrating about the axis Ax, 

 drawn perpendicularly through the plane of rotation passing through the centre 

 y. Let y be the centre of the centripetal forces taken without the fluid ; and 

 c the centre taken in the section towards which the parts of the fluid are 

 attracted. 



That the parts of the fluid may remain in equilibrio, the weight of any co- 

 lumn CD by gravitating towards y, and towards c, and by a centrifugal force, 

 should remain every where the same. 



Let then the gravity at a towards y be given, and = tt, the gravity at a to- 

 3x2 



