340 Mr. Ivory's Remarks on the Theory 



and consequently in equilibrio; and we may inquire what 

 change of form will ensue in consequence of a rotatory motion 

 causing a centrifugal force very small in proportion to the 

 gravity. The problem is considered in this view in the Prin- 

 cipia ; but no investigation is given of the nature of the oblate 

 figures that will have their particles in equilibrio by the action 

 of the attractive and centrifugal forces. Newton tacitly as- 

 sumes that the fluid sphere, in the nascent change of its form, 

 will become a spheroid such as is generated by the revolution 

 of an ellipse about the less axis. The meridians of the qui- 

 escent sphere are thus, in the revolving figures, changed into 

 ellipses having the greater axis in the equator. But whether 

 this assumption was made merely because the ellipse is the 

 most simple of oval figures, or for some other reasons, it 

 would be in vain to inquire. 



Supposing therefore that the oblate figures caused by the 

 centrifugal force are elliptical spheroids, we have still to de- 

 termine the relation between the protuberance at the equator, 

 and the observed velocity of rotation. Now this research is 

 greatly assisted by the consideration that the spheroids are 

 very little different from spheres. For, according to the 

 general law that regulates the small variations of mathema- 

 tical quantities, the centrifugal force at the equator and the dif- 

 ference between the equatorial and the polar diameters, will al- 

 ways have the same proportions to the gravity and the polar 

 axis, so long as we can neglect the squares and other powers 

 of the first two quantities. It is sufficient therefore to deter- 

 mine what these proportions are in some given figure. New- 

 ton takes the case of the spheroid that has the polar axis 

 equal to 100 parts, and that of the equator to 101 of the same 

 parts ; and he computes that the gravity of a particle placed 

 at the pole, is to its gravity at the equator as 501 to 500. 

 He next supposes two columns of the fluid reaching from the 

 centre of the spheroid, one to the pole, and the other to the 

 equator ; and as any two particles similarly placed in these 

 columns will have their gravities in the constant proportion 

 of 501 to 500, it follows that the total weights will be to one 

 another as 501 x 100 to 500 x 101, or as 501 to 505. Where- 

 fore, if we suppose the spheroid at rest, the equatorial will 

 preponderate the polar column ; but, if we suppose a rotatory 

 velocity sufficient to diminish the gravity of the particles in 

 the equatorial column by its g-^ part, the weights of the two 

 columns will just balance one another, and the revolving 

 spheroid will be in equilibrio. Thus, when the protuberance 

 at the equator is t ^q of the polar semi-axis; the centrifugal 

 force requisite to the equilibrium is T $ c of the gravity at the 



equator^ 



