328 



KOTE IV. 



5. THE problem to determine the ellipticity of a planet 

 considered as heterogeneous, is by no means an easy one. It 

 certainly was beyond the powers of an age when the laws 

 that govern the equilibrium of fluids were almost unknown. 

 Newton determines the form of equilibrium from the con 

 dition that the weights of all columns of fluid, from the 

 centre to the surface, must be equal. This, however, is 

 not sufficient for equilibrium. Huygens added afterwards 

 another condition, that the form of the surface must 

 always cut perpendicularly the direction of the resultant 

 force. But even these two conditions together are not 

 sufficient. Clairaut (Figure de la Terre, Chap. III.) gave 

 an instance in which, under a particular law of gravity, 

 the particles of fluid could be so arranged that both these 

 conditions were satisfied; yet he also showed that, so far 

 from the fluid being in equilibrium, it was actually impos 

 sible for any fluid to rest in equilibrium under the action 

 of these forces. It was Clairaut who first investigated 

 all the necessary conditions of equilibrium, and showed 

 that both the principles hitherto used were included in the 

 one he proposed. His famous work, l( Theorie de la 

 Figure de la Terre&quot; was published in 1743, and in it he 

 applied his theory to determine the form of the earth 

 considered as heterogeneous. Very little has been effected 

 in this subject since his time. The form of the investi 

 gation has been changed, but all the results remain essen 

 tially the same. The form of the earth, whatever it may 

 be, must consist of &quot; level &quot; strata of equal density, of 

 which the surface is one. Clairaut assumes all these to be 



