Mr. Ivory »« the Theory uj the Figure oj' the Earth. 'iiS 



ing upoa a particle in the outer surface be perpendicular to 

 that surface ; 2dly, That any interior body of the fluid bounded 

 by a level surface be in eqidlibrio with respect to the attrac- 

 tion of all the exterior matter. The latter condition cannot 

 be fulfilled unless every level stratum possess such a figure 

 as to attract all particles in the inside with equal force in op- 

 posite directions. 



In the paper alluded to, the subject is examined at great 

 length. The inadvertence is pointed out both in the original 

 investigation of Clairaut, and in the usual theory of fluids, 

 founded on the principle of equal pressure in all directions. 

 It is proved that both these methods lead to the conditions of 

 equilibrium above mentioned, when no part of the forces in 

 action is left out. The case of a homogeneous fluid revolving 

 upon an axis, and composed of particles which attract one an- 

 other inversely as the square of the distance, is next consi- 

 dered ; and the conditions of equilibrium are investigated by 

 means of particular considerations without the aid of any ge- 

 neral theory. — Thinking that it may gratify some of the readers 

 of the Philosophical Magazine, to whom the progress of the 

 philosophy of Newton is not altogether indifferent, I have 

 subjoined the last-mentioned investigation, which does jiot re- 

 quire algebraical calculations. It is contained in the two fol- 

 lowing propositions. 



Prop. I. — If a homogeneous fluid body revolving upon an 

 axis be in equilibria by the attraction of its particles in the in- 

 verse proportion of the square of the distance; any other mass 

 of the same fluid having a similar figure, and revolving upon 

 an axis similarly placed,will likewise be in equilibrio, supposing 

 that its particles attract one another by the same law. 



Suppose that the two bodies are similarly divided into the 

 same indefinitely great number of molecules, of which dm and 

 dm' are any two situated alike, and therefore having their 

 volumes aiul quantities of matter proportional to the volumes 

 and quantities of matter of the two whole bodies. Take also 

 any two points similarly situated, either witiiin both bodies 

 or in their outer surfaces; and lety'and /' denote the respec- 

 tive distances of the two points from the molecules dm and 

 d m\ and ;• and r' their distances from the two axes of revo- 

 lution. 



The forces with which the molecules dm and dm' attract 

 two e(|ual particles of matter placed in the assumed points 



are porportional to -jp and — - ; and, in these fractions, the 



niniu;rators being proportional to the cubes, and the dcnomi- 

 i)at(»rs to the s(juares of any two homologous lines of the re- 



J I h '^ spcctive 



