326 Mr. Ivory on the Figure of the Planets, 8fc. 



it is impossible by means of that equation to deduce from the 

 sphere any other figure beside the elliptical spheroid. If we 

 set out from any other initial figui'e but the sphere, or, more 

 generally, from any other figure but one of the elliptical 

 spheroids into which the same mass of fluid may be moulded, 

 it will be impossible to obtain any solution of the problem of 

 a homogeneous planet by means of the equation (1) alone. 



The rationale of the analytical process of Legendre is un- 

 doubtedly such as we have just explained. It is founded on 

 the connection that necessarily subsists between all the ellip- 

 tical spheroids with which the equilibrium of the same homo- 

 geneous fluid mass is possible; a connection which enables 

 the analyst to deduce all the related figures from any one by 

 means of their common equation (1). The success of the 

 investigation depends upon a mathematical relation of the 

 figures, and not upon the conditions of equilibrium which are 

 inherent in the figures themselves. No just objection can, 

 therefore, be urged against the theory we have laid down, 

 from the process of Legendre, w'hen the principles of it are 

 rightly understood ; besides, the result obtained by it is no- 

 wise inconsistent with the conclusions at which we have ar- 

 rived. But the method of Legendre is very imperfect. It 

 necessarily supposes that a is a small fraction ; because a. is 

 contained within a certain limit in all the related spheroids. 

 Although it is proved that the analytical operations are possi- 

 ble, yet it is found that they cannot be practically executed 

 after the first step. It has no pretensions to be called a 

 solution a priori deduced solely from the hydrostatical laws 

 of equilibrium. In all these respects it is inferior to the view 

 we have taken of the problem ; which ascending to the first 

 principles of the research, derives, by a general analysis, the 

 absolute conditions which are necessary and sufficient to the 

 solution ; and by this means reduces the mathematical deter- 

 mination of the figure to the utmost simplicity of which it is 

 capable. 



On another occasion I will show that the equilibrium of a 

 homogeneous mass of fluid furnishes the only key to the ge- 

 neral solution of the problem of the figure of the planets in a 

 fluid state, and supposed to consist of heterogeneous strata ; a 

 problem which has not yet been touched upon, at least with 

 success, except by Claii-aut, whose investigation includes only 

 the first power of the oblateness. 

 Oct. 11, 1827. James Ivory. 



LIV. On 



