of the Figure of the Earth. 347 



fluid consisting of attracting particles. The full conditions 

 requisite to the equilibrium of such a fluid mass are these : 

 1°. The resultant of the accelerating forces must be perpen- 

 dicular to the outer surface, and the differential equation of 

 the same surface must be an exact fluxion ; 2°. Every level 

 stratum must be possessed of such a figure as to attract all 

 particles in the inside with equal forces in opposite directions. 



Let us now consider the equilibrium of a fluid mass diffe- 

 rently ; in the method of Euler, and as it is treated in most of 

 the elementary works. For this purpose we must find the 

 conditions requisite to the equilibrium of a rectangular paral- 

 lelopiped of the fluid placed any where in the mass. The 

 forces in action are: 1°. the pressures upon the six faces 

 tending to compress the fluid into a less space ; 2°. the acce- 

 lerating forces acting upon the particles of the parallelopiped. 

 If the latter forces be resolved into three sums perpendicular 

 to every two faces of the parallelopiped ; it is obvious that 

 each sum must, in the case of an equilibrium, be equal and 

 opposite to the difference of the pressures upon the same two 

 faces. Three separate equations are thus formed; and by 

 combining them, we deduce the value of the differential ol 

 the pressure ; and again, if we suppose the pressure constant, 

 we obtain the equation of the level surfaces. Nothing can be 

 clearer or more simple than this procedure, when there is no 

 attraction between the particles. In this case it is unquestion- 

 able that all the forces in action are taken into account, and 

 no objection can be made to the accuracy of the result. But 

 when the particles attract one another, some reflection will 

 show that there is an omission. In estimating the accelerating 

 forces, the attraction of the exterior matter upon the paral- 

 lelopiped is alone considered, while the attraction of the par- 

 ticles of the parallelopiped upon the exterior matter is neg- 

 lected. Although it is supposed that the parallelopiped is in- 

 definitely small, yet, as the attraction of its particles is ex- 

 tended to all the fluid mass, an accumulated force is produced 

 comparable to the pressure, and which must not be omitted 

 in adjusting the equilibrium. When all the forces acting 

 upon the parallelopiped ; both those extrinsic to its own mat- 

 ter, and those inherent in its particles ; are taken into account, 

 the same conditions of equilibrium will be obtained that have 

 already been found by the former investigation. 



The proofs of the new theory of the equilibrium of a fluid 

 consisting of attracting particles, are fully detailed in a pap< ir 

 sent to the Royal Society in November last, and which will 

 appear in their Transactions for the present year. Having 



X x 2 obtained 



