274 Mr. Ivory on an Article in the Bulletin ties Sciences. 



rived from the surface of the planet by the same determinate 

 construction, it only remains to find the figure that will re^ 

 concile them with the forces that prevail in the interior of the 

 fluid. It will be found that the condition, R"= 0, is indis- 

 pensably required ; which limits the figure of equilibrium to 

 the elliptical spheroid. 



Whenever the resultant of the forces R' and R" is perpen- 

 dicular to the surface of A, that surface will be one of equa- 

 ble pressure. Now there are two ways in which this may 

 happen : each force may be separately perpendicular to the 

 surface, as when A is similar to the planet and similarly po- 

 sited about the centre of gravity, which requires the condi- 

 tion R" = 0; or the resultant may be perpendicular to the 

 surface, although both the forces be oblique to it. When the 

 forces are computed, the analytical investigation will be found 

 to bring out the same equation of the surface of A in both 

 cases; but this equation admits of two solutions, which deter- 

 mine the two surfaces of equable pressure. Thus in every el- 

 liptical spheroid in equilibrio there are two sorts of surfaces 

 of equable pressure, both elliptical but differing in their figure. 

 Further, the planet being in equilibrio, any interior body, as 

 A, bounded by a surface of equable pressure, will be in equi- 

 libria separately, the stratum C — A being taken away; and 

 hence we learn the reason why, the data of the problem being 

 the same, there are two, and only two, different figures of 

 equilibrium. 



Clairaut's theory may be reconciled with one of the sets of 

 surfaces of equable pi'essure, because the force R" which that 

 theory leaves out, disappears on account of the condition 

 R" = : but the co-existence of two different surfaces of 

 equable pressure in the same mass of fluid in equilibi-io is en- 

 tirely incompatible with the theory in question. 



What has been said proves that all the researches founded 

 on Clairaut's theory, which makes the perpendicularity of 

 gravity to the surface of the planet the sole condition of equi- 

 librium, must fail in leading to an exact solution of the pro- 

 blem ; although in some cases they may bring out an approxi- 

 mation. Of all the possible figures possessing the property 

 mentioned, the oblate cllij)tical spheroid is the only one which 

 at the same time fulfils all the conditions that are true in na- 

 ture. On the whole, there is, in reality, no part of the theory 

 of the equilibrium of a planet perfectly established on exact 

 principles except the synthetic demonstration given long ago 

 by Maclaurin with all the elegance and rigour of the ancient 

 geometry. These observations may suffice at present on this 

 subject, which requires to be reviewed and treated anew from 

 first principles. The 



