166 Mr. Ivory on the Figure of the Planets^ 



The foregoing propositions, which have been deduced by 

 analytical reasoning from the single hypothesis that there is an 

 equilibrium, contain essential properties, without which the 

 equilibrium cannot subsist in any possible figure of the fluid 

 mass. Another property is now to be added, no less essential, 

 no less independent of all considerations relative to a particu- 

 lar figure, and equivalent to the new condition employed for 

 finding the figure of a homogeneous planet, to which M. Pois- 

 son has objected. 



Prop. 4. — Suppose that a homogeneous mass of fluid, re- 

 volving about an axis which passes through the centre of gra- 

 vity, is in equilibrio by the attraction of its particles in the in- 

 verse proportion of the square of the distance ; take any point 

 in the interior of the fluid, the distance of which from the 

 centre of gravity is equal to r, and let V(r) denote the sum of 

 all the molecules of the whole mass divided by their respec- 

 tive distances from the assumed point, and V'(/) the like sum 

 extending only to all the molecules within the level surface 

 passing through the same point : then is V(r) — V'(r) a con- 

 stant quantity for all points of the same level surface. 



Let <p denote the centrifugal force at the distance ' from 

 the axis of rotation, and S the angle which r makes with the 

 same axis : then, as is well known to all geometers, thee qua- 

 tion of the interior level surface will be, 



V(r) + <|5 x4sin^6 = C, (1) 



C being constant for all points of the same level surface, and 

 varying from one level surface to another. But, according to 

 the second and third propositions, the portion of fluid bounded 

 by the interior level surface would be in equilibrio without 

 any change of its figure, if it revolved by itself; in which case 

 the equation of the surface would be, 



V'(r) + (}> X -JsinM = C. (2) 



Wherefore, by subtracting the two equations, we get, 



V(,-) - V'(r) = C - C, (3) ' 



which is the property to be proved, because C and C are the 

 same for all points of the same level surface. 



The property just investigated and the equation of the outer 

 surface of the fluid are together sufficient for determining the 

 figure of a homogeneous planet. But without the property 

 mentioned, the equihbrium would not take place. For nothing 

 followsfrom the equation of the outer surfaceof the fluid alone, 

 except that the equation (2) would be true of all the interior 

 surfaces similar to the outer one and similarly posited about 



the 



