500 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



pressure at all the points of the surface will he n x^p; and the equation of the sur- 

 face being 



C — nX lp = <p{oD,y,z), 



if jo = n X ^p, we shall have 



C^p=z<p{x,i/,z), 



which equation ascertains the pressure at any point {xy%), and determines the sur- 

 face containing all the points at which the same pressure prevails. This agrees with 

 what was investigated in No. 3. 



The interior surfaces at all the points of which the pressure is constant have been 

 called level surfaces ; and a stratum of the fluid lying between two level surfaces is 

 called a level stratum. 



A property common to all the level surfaces, and to the upper surface of the fluid, 



consists in this, that the resultant of the forces acting upon a particle contained in 



any of these surfaces is directed perpendicularly towards it. Take two points, {<ry z) 



and {cc ■\-lx,y -\-ly,z •\-l z), infinitely near one another in the surface of which the 



equation is 



C—p=z(p{x,y,z)i 



and put Is for the short line between the two points ; by diffbrentiating, C — p being 



constant, we get 



d <^ d X d <p dy d <p dz 



d x' ds * d 1/' ds * dz ' ds ' 



or, which is equivalent, 



^ds ^^ -dJ-^^IJ-^' 



d x d u d s 

 Now -1—, -jj, -jj are the cosines of the angles which the directions of the forces make 



with the line ds: wherefore the algebraic expression in the last equation is the sum 

 of the partial forces which act in the direction of ds; and as this sum is equal to 

 zero in all positions of that line round the point (xy z), the forces will produce no 

 effect in the plane touching the curve surface, and will exert their whole action at 

 right angles to the surface. 



From what is here investigated, we may derive this general property : If the forces 

 X, Y, Z, which vary from point to point, be always perpendicular to a surface, they 

 must satisfy this equation, 



Xdx-\-Ydy-\-Zdz = 0, 

 the coordinates being made to vary in the surface. For if the equation be divided 

 by ds = tjdx^ -\- dy'^ -\- dz^, the result will be 



which expresses, as is shown above, that the whole action of the forces is perpendi- 

 cular to the surface. 



