FLUID PARTICLES. 77 



y 



or -^= a constant, which we may take for unity; therefore 



F= V. Resolving this force along the axes, 



X=V'f^-; whence X=fx, and so Y=fy, Z=fz. 



to is the increment of pressure = &p; multiplying the three 

 equations (1) by X, Y, Z } and adding, we get 



or putting d for S, 



dp = -Xdfe + Ydy + Zdz ............ . ..... (2), 



the equation of equilibrium of an homogeneous and incom- 

 pressible fluid, whose density is unity. 



An objector to Clairaut's reasoning might urge, that this 

 result, though certainly sufficient, was not shown to be neces- 

 sary : he might argue, that a way has been shown of building 

 up a fluid mass; but that it has not been proved that every 

 fluid mass is capable of resolution into the smaller masses, by 

 means of which alone Clairaut investigates the conditions of 

 equilibrium. Unless it be made a direct postulate, that every 

 fluid mass in equilibrio will continue in equilibrio, when the 

 part of it contained between the free surface and any level sur- 

 face is removed, it is difficult to see how this objection can be 

 met, except by showing that the property assigned by Huygens 

 to a free surface, viz. that the force is normal to it, belongs to 

 every surface of equal pressure, and that consequently Clairaut's 

 reasoning is in reality independent of any construction or reso- 

 lution of a fluid mass into successive strata. When we assert, 

 with Clairaut, that a fluid mass in equilibrium is not disturbed 

 by the addition of a stratum producing equal pressure, we imply 

 that the reaction produced at any point of the surface of A 9 by 

 the pressures exerted over the rest of the surface, i. e. the effect 

 of the transmitted pressures, is normal to it. For we know 

 that the forces at the surface are so ; and unless the inference 

 first stated is correct, there could be no equilibrium. It hence 

 appears, that Clairaut's axiom is equivalent to this Equable 

 pressure produces a reaction normal to the surface on which 

 it is applied. But if the force at a surface of equal pressure 

 were not normal to it, there could be no equilibrium, because 



