Mr. Ivory on the Figure of the Earth. 243 



may expect that it merits some regard from that learned body, 

 by which no important point of the philosophy of Newton can 

 ever be deemed indifferent or unfashionable. If new informa- 

 tion be wanted, I engage to produce demonstrations that will 

 bring the question to a perfect decision, as soon as I know 

 that a proper use will be made of them. This will only put 

 me to the trouble of transcribing part of what I have written. 



Srdly, Maclaurin demonstrated that a homogeneous planet 

 supposed fluid, is in equilibrium when it has the figure of an 

 oblate elliptical spheroid of revolution ; and lie investigated the 

 equation of the surface of the spheroid. D'Alembert, in ex- 

 amining this equation, discovered that it admits of more than 

 one solution ; and it is now well known that the number of 

 solutions is two, and no more. It is therefore a mathematical 

 deduction from the equation of Maclaurin, that the same mass 

 of fluid, revolving with the same angular velocity, will be in 

 equilibrium in two different figures. But there must be a 

 physical reason that determines tke number of figures of equi- 

 librium : and this reason must be a part of a solution of the 

 problem a jn-iori. Accordingly, in examining the forces in 

 action in the interior of the mass, which forces are entirely 

 omitted in the usual manner of solving the problem, I found 

 that two different sets of surfaces may be traced within the 

 fluid, each of which is possessed of the property of the level 

 surfaces in Clairaut's theory, that is, the intensity of pressure 

 is the same at all their points. The two sets of interior sur- 

 faces have different relations to the outer surface, and one set 

 only can properly be called level surfaces. The definition of 

 the level surfaces given by Clairaut is exact only in one par- 

 ticular case of the equilibrium of a homogeneous fluid entirely 

 at liberty ; and no other definition has ever been thought of 

 by any geometer. 



4-thly, When the difficulties respecting the equilibrium of 

 a homogeneous fluid are overcome, the same principles are 

 easily applied to a heterogeneous fluid. In this latter case the 

 equation of the surface cannot be found in a finite expression, 

 but it may be determined to any refiuired degree of approxi- 

 mation. Clairaut solved the problem long ago, retaining only 

 the first power of the ellipticity. I have already published in 

 this Journal*, a solution which takes in the second power of the 

 oblatcness, by a method which leaves no doubt respecting the 

 cquilibriuui of the fluid, and which requires no more than the 

 labour of calculation to extend it to any power of the oblateness. 



• I'liil. Mag. for July 182G, pp. 5 & G. The formula only is given, as the 

 analysis would have taken up too innch sjiace. IJnt I apprciienil tiiat a 

 forniula of this nature is suiricienL to bccin'e to its anilior the jjossession of 

 the inctliudof iiivcstii'ation, when il shall be in iiis power to publi^liit. 



2 I 2 The 



