498 



MR. IVORY ON THE EQUILIBRIUM OF A MASS OP 



Theorem. 

 If a body of homogeneous fluid at liberty have for the equation of its surface, 



C — <p{x,y,z), 



the mass will be in equilibrium, supposing that every particle (x y z) is urged by the 

 forces X, Y, Z, respectively parallel to the rectangular coordinates x, y, z, and equal 

 to the partial differential coefficients of (p {x, y, z), that is, 



dx ' dy ' dz 



The origin of the coordinates being placed at the centre of gravity of the mass, the 

 theorem must be supposed to assume further, that the expressions of the forces, that 

 is, the differential coefficients of <p {x, y, z), vanish when the coordinates are all equal 

 to zero ; for without this condition the equilibrium of the mass of fluid would be im- 

 possible. From this it follows that the value of (p {x, y, z) will increase continually 

 as the point (;r y z) recedes from the centre and approaches the surface of the fluid 

 on any side. If C° denote the value of <p {x, y, z) at the centre of gravity, C — C° will 

 be the whole increase in varying from that point to the surface of the fluid ; and as 

 every gradation of magnitude is passed through, an interior surface may be found that 

 will satisfy the equation 



C = (p {x, y, z), 



provided C be any quantity between the limits C and C°. Wherefore if C — C° be 

 divided into an infinite number of elementary portions, each equal to I p, there will 

 exist a series of curve surfaces gradually contracting in their dimensions round the 

 centre, and the last containing a drop of fluid, which may be as small as we please ; 

 of which successive curve surfaces, beginning with the upper surface of the mass, 

 these are the respective equations : 



C = (p{x,y,z),ovC = (p, 



Q-lp = (p, 



C — 2^j» = 9, 



C — '6lp = (p, &c. 



From A in the upper surface draw A A' perpendicular 



to the surface next below ; put A: = A A', the thickness 



of the stratum ; and let w denote any infinitely small 



portion of the curve surface at A' ; then k X w will be 



the portion of the stratum insisting on the small surface w. 



The coordinates of the point A' being x, y, z, the forces in 



action and respectively parallel to the coordinates will be 



d <p d<p d <p 

 dx^ dy^ dz ' 



