38 Mr. Ivory on the Equilibrium of Fluids. 



determine the position of the jwint of action, these coordinates 

 being considered as unrelated and. independent quaiitities. To 

 prove this, take any two points of tiie mass, and let a com- 

 munication be made between them by a canal of any figure : 

 if to the pressure of the fluid on the orifice of the canal at 

 one end, we add the effbrt of the fluid contained in the canal, 

 caused by all the forces that urge the particles in the direc- 

 tion of the canal, the result will counterbalance the pressure 

 on the orifice at the other end ; wherefore, in a fluid at rest, 

 the effort of every canal, of whatever figure, will be the same, 

 provided the extreme orifices ai'e the same. Now it is easy 

 to ascertain that this property will be fulfilled when the sort 

 of function described in the theorem stands for the pi'essure ; 

 and an application of the method of variations will show that 

 no other sort of function will answer the same end. 



The theorem being demonstrated, let X, Y, Z denote the 

 accelerating forces that urge a particle of the fluid in the re- 

 spective directions of its coordinates, .r, y, z ; the differential 

 of the pressure will be 



:^da: + Ydy + Zdz: _ _ (a.) 



and, as the fluid is at rest, this must be the differential of a 

 function of a certain kind, by the theorem ; and, on the other 

 hand, if it be an exact differential, it can be derived only 

 from a function of the sort mentioned. Wherefore, in a fluid 

 in equilibrium, the expression (a.) is always an exact diffe- 

 rential, and it is zero at the upper surface, because the press- 

 ure is constant at that surface. 



The foregoing reasoning holds, whatever modification, or 

 assumption, is superadded to the notion of fluidity. The 

 mathematical laws of the equilibrium of fluids are thus placed 

 on the broadest foundation. They are sufficient for deter- 

 mining the figure of equilibrium when the integral of (a.) is 

 an explicit function of the coordinates; but they are not suffi- 

 cient when the same integral is only an implicit function, that 

 is, when the forces in action are derived from different 

 sources, and are independent on one another ; in which case, 

 as common sense dictates, recourse must be had to the pecu- 

 liar circumstances of the problem in order to complete the 

 solution. 



It deserves to be noticed that the theorem is true, and the 

 expression (a.) is an exact differential, in a fluid at rest, but 

 not in one in a state of motion ; which marks a distinction be- 

 tween the mathematical laws that govern the two cases : yet 

 it is usual to deduce the motion of fluids from their equili- 

 brium. 



December 17, 1838. JamKS Ivory. 



