164- Mr. Ivory on the Figure of the Planets, 



the least, it is very doubtful whether it can possibly be accom- 

 plished. 



Being dissatisfied with the usual equations for finding the 

 figure of a homogeneous planet in a fluid state, I have sought 

 to deduce a solution of the problem by a strict analysis, with- 

 out admitting any gratuitous assumption. By analysis I do 

 not here mean an algebraic calculation ; I understand by it a 

 process of reasoning such as the ancient geometers employed 

 to derive the construction of a geometrical problem from the 

 conditions to be fulfilled. In the following investigation I con- 

 fine my attention to a homogeneous planet in a fluid state ; 

 that is, to the equilibrium of a fluid mass of uniform density, 

 the particles attracting one another inversely as the square of 

 the distance, and being urged by a centrifugal force caused 

 by rotation about an axis that passes through the centre of 

 gravity ; and in order that the train of reasoning may be more 

 easily examined, I divide it in distinct propositions. 



Prop. 1. — If two particles be similarly placed, in two bodies, 

 exactly similar in their figure and composed of the same ho- 

 mogeneous matter ; the attractive forces of the bodies upon 

 the particles will act in similar directions, and will be propor- 

 tional to the linear dimensions of the bodies. 



Particles here mean infinitely small portions of the two 

 bodies proportional to the whole masses. The proposition is 

 proved, Prin. Math. lib. i. Prop. 87.— Maclaurin's Fltixions, 

 § 629. 



Prop. 2. — If a homogeneous mass of fluid revolving upon 

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

 inverse proportion of the square of the distance ; any other 

 mass of the same fluid having a similar figfure and revolving 

 with the same rotatory velocity about an axis similarly placed, 

 will likewise be in equilibrio, supposing that its particles at- 

 tract one another by the same law. 



Take two particles similarly situated in the two bodies. By 

 Prop. 1. the resultants of the attractive forces acting on the 

 particles have similar directions, and are proportional to the 

 linear dimensions of the bodies. Further, the centriftigal forces 

 urging the particles to recede from the axes of rotation, are 

 proportional to the respective distances from those axes, that 

 is, to the linear dimensions of the bodies. Wherefore the 

 joint effect of all the forces is to urge the particles in similar 

 directions with intensities proportional to the linear dimen- 

 sions of the bodies. And as the same thing is true of all par- 

 ticles similarly situated in the two bodies, it follows that if 

 there be an equilibrium in one case, diere will likewise be an 



equilibrium 



