322 Mr. Ivory oji the Figure of the Planets, 



which it is the expression, that determines the figure of a ho- 

 mogeneous planet in equilihrio, to be an elliptical spheroid 

 exclusively of all other figures ; and the equation of the outer 

 surface ascertains the rotatory velocity I'equisite to maintain a 

 given oblateness of the planet. 



In the mode of investigation we have followed, every point 

 has been strictly demonstrated from principles universally ad- 

 mitted, and to which no objection can be made. The con- 

 clusions at which we have arrived, rest on their own evidence, 

 and are independent of what it is usual to lay down systema- 

 tically respecting the equilibrium of a homogeneous fluid. 

 We are under no necessity of entering upon any discussion 

 of the received doctrine for the purpose of removing difficul- 

 ties, or of seeking new arguments in favour of what has been 

 rigorously proved : but it will be satisfactory, and it cannot 

 but contribute to remove some erroneous notions prevalent 

 upon a difficult subject, which has been treated by all authors 

 in a manner too general and devoid of precision, if we show, 

 by a careful examination, that the theory which has prevailed 

 unquestioned from the time of Clairaut and Euler, affords 

 no solid ground for the objections which have been urged 

 against the solution we have pi'oposed. 



The equilibrium of a homogeneous planet in a fluid state, 

 to which we confine our attention for the present, requires 

 nothing more, according to the usual theory, than that the 

 force of gravity be every where perpendicular to its surface. 

 When this single condition has been deduced from the laws 

 of hydrostatics, the problem, it is said, is brought within the 

 dominion of the mathematics, and has no further dependence 

 on physical considerations for its solution. But if we inquire 

 into the grounds on which all this is asserted, we shall find 

 that they are not altogether sure and satisfactory. No clear 

 demonstration has ever been given that the equilibrium is a 

 necessary consequence of the perpendicularity of gravity to 

 the surface of the planet. This is no doubt always an essen- 

 tial condition ; and, when the accelerating forces urging the 

 particles of the fluid are explicitly given, it is sufficient alone 

 for investigating all the circumstances of the equilibrium. But 

 the mutual attraction of the particles introduces new forces, 

 which vary with the figure assumed by the mass of fluid, and 

 modify the solution of the problem. The hydrostatical theory 

 as applied to a homogeneous planet is defective in the evidence 

 of its fundamental principles ; and it has remained steril and in- 

 effective. The only application that has been made of the hydro- 

 statical theory is when the planet in equilihrio is very little dif- 

 ferent 



