HOMOGENEOUS FLUID AT LIBERTY. 523 



the particle P is not infinitely little ; it is expressed by the difference of two definite 

 integrals ; and, however small in degree the pressures urging P on different sides 

 may be supposed, yet, if they be unequal, the particle must move in the direction in 

 which the force is greatest. By omitting the attraction of the stratum, the procedure 

 of Clairaut is evidently defective, and applicable only to such fluids as consist of par- 

 ticles that have no action upon another. 



But the investigation of Clairaut, although limited as it is laid down by the 

 author, when it is stated with all the generality of which it is susceptible, will be 

 found on due reflection to contain the only true and satisfactory principle of the 

 equilibrium of a mass of fluid at liberty =*<=. To render it perfectly general, nothing is 

 wanting but to take into account all the forces necessary to complete the equilibrium 

 at every separate stage of the procedure. The original mass H K I being supposed 

 in equilibrium, the stratum on I must be adjusted as Clairaut directs, so as to exert 

 a constant pressure ; but a new condition must be added, that the body of fluid H K I 

 be in equilibrium by the attraction of the stratum, that is, the pressures caused in the 

 mass H K I by the attraction of the stratum, must urge every particle of it with the 

 same intensity on all sides. When these conditions are fulfilled, the body of fluid, 

 consisting of H K I and the stratum onl, will be in equilibrium, and its upper sur- 

 face will be stable as was that of H K I, and capable of supporting additional strata. 

 A new mass in equilibrium will be formed by adding a second stratum, so as to fulfill 

 the same conditions as the first, that is, it nmst press with the same intensity at all 

 points of the surface below it, and its attraction must have no power to move the 

 particles contained within it. Continuing the same procedure and adding more strata 

 indefinitely, a body of fluid of any dimensions will be formed, which is in equilibrium, 

 all the forces in action being taken in account. 



If we now examine a mass of fluid constructed by the foregoing process, so as to 

 be in equilibrium, it is obvious that all the successive surfaces are deduced in the 

 same manner from the forces acting on the particles contained in them. If the forces 

 be explicit functions of the coordinates of their points of action, the condition that 

 every surface must be pressed with the same intensity at all its points, determines the 

 general equation of all the surfaces, nothing varying from one surface to another but 

 the magnitude of pressure, as in the theorem in No. 5. The upper surface contains 

 all the points of the fluid at which there is no pressure, and its equation alone ascer- 

 tains the figure of equilibrium. This is the theory of Clairaut in its full extent, and 

 it is comprised in the theorem alluded to : but if the forces in action are not explicit 

 functions of the coordinates, but depend upon the very figure to be investigated, the 

 condition that the pressure must be constant in every successive surface, leads to an 



* It is obvious that all the steps of Clairaut's procedure must be perfectly similar. As the central body 

 H K I is supposed in equilibrium, so the addition of every stratum must produce a body in equilibrium, all the 

 causes capable of moving a particle being taken into account ; if not, the process cannot be continued, or will 

 fall into error. 



