'2i'2 Mr. Ivory n/i llie 'I'heon/ of' tin: Figure of the Earth. 



The result of Clairaut's researches are thus concisely stated 

 in another passage of the same author: 



" Les conditions de I'equiiibre d'une masse fluide sont done : 

 1°, que la direction de la pesanteur soit perpendiculaire a 

 chaqiie point de la surface exterieure ; 2°, que dans I'interieur 

 de la masse, les directions de la pesanteur de chaque mole- 

 cule, soient perjiendiculaire a la surface des couches de den- 

 sitc constante. Connne on pent dans I'interieur d'une masse 

 homogcne, prendre telles couches cjue Ton veut, pour couches 

 de densite constante; — la seconde des deux conditions prece- 

 dentes de I'equilibi-e est toujours satisfaite, et il suffit pour 

 I'equiiibre que la premiere soit remplie ; c'est-a-dire que la 

 resultante de toutes les forces qui animent chaque molecule 

 de la surface exterieure, soit perpendiculaire a cette surface." 

 — Mt'c. Celeste, torn. 2'^S liv. 3"% p. 64. 



Confining our attention to the case of a homogeneous fluid, 

 the conditions of equilibrium are extremely simple. The outer 

 surface must be defined by an equation between three inde- 

 pendent co-ordinates ; and the resultant of all the forces act- 

 ing at every point of the same surface must be perpendicular 

 to it. Being in possession of a theory so little complicated, 

 we should expect to be able to deduce from it the beautiful 

 discovery of Maclaurin ; and to prove, a priori, that the ellip- 

 tical spheroid is one figui'e of equilibrium, although, perhaps, 

 not the only one. No geometer has succeeded in this investi- 

 gation. It may be said with truth, that the received theory of 

 the equilibrium of fluids is of little use in the question of the 

 figure of the planets; for the small change induced upon a 

 fluid sphere by the action of a centrifugal force is a mathe- 

 matical conse(juence of the perpendicularity of gravity to the 

 outer surface. 



In a paper inserted in the Philosophical Transactions for 

 1824', I have shown that there is an inadvertency in the theory 

 of equilibrium as it is delivered by Ciairaut, which is the true 

 cause of all the difficulty and embarrassment that has oc- 

 curred in the determination of the figure of the planets. The 

 conditions of equilibrium are rightly assigned when there is 

 no mutual attraction between the particles ; or, to speak more 

 precisely, when all the forces that act upon a particle are in- 

 dependent of the matter without the level surface in which the 

 particle is placed. M'hen the particles are endowed with at- 

 tractive powers, Clairaut's theory is insufficient ; and it lie- 

 comes necessary to add a nev/ condition in order to ensure 

 the ecjuilibrium. In the case of a homogeneous fluid com- 

 posed of particles that mutually attract one another, the equi- 

 librium requires, 1st, That the resultant of all the forces act- 

 ing 



