in reply to M. Poisson. 325 



gendre is insufficient to prove the equilibrium of a homoge- 

 neous planet. The only question that remains to be consi- 

 dered, in order to enable us to form a correct judgement of 

 his investigation, is to inquire whether the figure determined 

 by the equations (1) and (2) be necessarily an elliptical sphe- 

 roid. If this shall be found to be the case, it must be occa- 

 sioned by the influence of some principle that has escaped 

 notice. 



A little reflection will soon convince us that the value of j/ 

 deduced from the equations (1) and (2) can possibly be no 

 other than the radius of an elliptical spheroid expanded in a 

 series. This is in reality a mathematical consequence of the 

 discovery of Maclaurin. An elliptical spheroid of any oblate- 

 ness will be in equilibrio, as was proved by that geometer, 

 when it revolves with a proper rotatoiy velocity. The same 

 mass of fluid may, therefore, assume every variety of the form 

 mentioned, from the sphere to the most extreme oblateness, 

 and still be in equilibrio in every case, provided the rotation 

 be adapted to the figure. Now the equation ( 1 ) is common 

 to all the spheroids i7i equilibrio, expressing generally the re- 

 lation between the oblateness and the centrifugal force. It 

 follows, therefore, from the received principles of analysis, that 

 if we set out from any one of the spheroids in question, we 

 may thence find the whole suit of related figures by means of 

 their common equation. 



The investigation of Legendre falls exactly under the de- 

 scription just given. The calculation begins at the sphere, 

 which is one of the figui'es that fulfill all the conditions of the 

 equilibrium of a homogeneous fluid. It is no doubt an ex- 

 treme case, the oblateness and centrifugal force being both 

 evanescent; but, for this very reason, it is the most favourable 

 point to set out from. In the sphere we have a = ; and if 

 we suppose that a is a very small fraction, the square of which 

 may be neglected, the resulting figure is found to be an ellip- 

 tical spheroid. If we now push the calculation to include the 

 powers of «, it is proved at least that the series for j/ has only 

 one form, and consequently that there is only one solution of 

 the problem. But we know, from what Maclaui'in has proved, 

 that, when a is very small, there is one elliptical spheroid, 

 and only one, depending upon the quantities a and «, which 

 will satisfy the equation ( 1 ) ; whence it follows that the series 

 for y can be no other than the radius of the spheroid ex- 

 panded in a series. We have here made use of M. Poisson's 

 argument, but. we have placed it on its proper foundation. It 

 is not because there must be an equilibrium on account of the 

 c({uation { 1 ), that the two figures must coincide ; but because 



it 



