Mr. Ivory on the Theory of the Fignre of the Earth. 245 



intercepting very thin strata of the fluid between them. Then 

 the whole fluid body K S T, and the part of it bounded by 

 the surface O P Q, are similar to one another in their fitrure ; 

 and they revolve about the common axiy A B, which cuts them 

 both similarly ; wherefore, because the first body is in equUi^ 

 brio, the latter will also be iti cquilihrio, supposing that it re- 

 volves by itself, the exterior stratum being taken away or an- 

 nihilated *. And, because the body O P Q is in equilibrio 

 when it revolves by itself, the gravitation at its surface, or 

 the resultant of the attraction of its particles and the centrifu- 

 gal force, will at every point be perpendicular to that surface. 

 Wherefore every particle of the fluid contained in the stratum 

 between the surfaces R S T and O P Q will be urged, by the 

 gravitation at the latter surface, perpendicularly towards it. 

 But the matter of the same stratum, besides pressing upon the 

 surface O PQ, likewise attracts all the particles included within 

 that surface. And the pressure and the attraction we have 

 mentioned, are all the forces which the stratum exerts upon 

 the fluid body O P Q Now the body O P Q is in equilibrio 

 in two different states : namely, when it is a part of the whole 

 body R S T ; and when it revolves by itself, the exterior mat- 

 ter being taken away : it must therefore be in equilibrio with 

 respect to the difference of the forces which act upon it in the 

 two states of equilibrium ; that is, it must be in equilibrio by 

 the pressure and attraction of the exterior stratum. And be- 

 cause the gravitation is perpendicular to the outer surface 

 R S T, and likewise to the interior surface O P Q, indefinitely 

 near the outer one, this latter must be a level surface, and the 

 pressure of the exterior stratum upon any given space assumed 

 in it must be eveiy where the samef. Wherefore the body 

 O P Q will be iti equilibrio by the pressui'e of the exterior 

 stratum acting separately; and consequently it must likewise 

 be in equilibrio by the attraction which the same stratum ex- 

 erts upon it. The stratum between the surfaces R S T and 



• Prop. I. 



\ Let the equation of the outer surface be 

 ?) = C: 

 then the equation of another surface indefinitely near it, which is perpen- 

 dicular to all the lines to which the first is perpendicular, will be 



<p = C — 3 C, 

 3C being a small variation of C. Let Ic denote the small distance of the 

 two surfaces at any point, and p the gravitation at cither extremity of k ; 

 then it is easy to prove that 



k X. p = oC. 

 Now k X p, or the product of the gravitation multiplied by the quantity 

 of matter, is the pressure ; and the c(|uation shows that the pressure upon 

 a given space is the same over all the surface. 



OPQ 



