ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 7 



and the aggregate or sum of these products becomes 

 p b X ba + d X dc+fxfe + h X hg -\- k X ki + mX ml, 



where p denotes the sum of the computed pressures. 

 Now, it is manifest, from what we have demonstrated in Proposition 

 (A), respecting the centre of gravity of a system of bodies, that 



The sum of the products, arising from multiplying each 

 body into its distance from a certain plane given in position, 

 is equal to the sum of the bodies, drawn into the distance of 

 their common centre of gravity from that plane. 



Let therefore, the particles b, d,f, h, k and m be considered as a 

 system of very minute bodies, and let the surface of the fluid denote 

 the plane given in position, to which the system is referred ; then, if 

 G be the place of the common centre of gravity of that system, put 

 71 G == 3, and we shall obtain 



8 (b+d+f+h+k+m) = b.ba+d.dc+f.fe+h . hg+k . ki+m.ml. 



But we have seen above, that the sum of the products on the right 

 hand side of the equation, expresses the aggregate pressure on the 

 several points of the containing surface, to which the present step of 

 the inquiry refers, and that pressure we have briefly represented by the 

 symbol p ; therefore we have 



and this expression implies, that the pressure exerted by a fluid, on 

 any number of points of the surface that contains it, 



Is equal to the sum of the points, drawn into, the perpendi- 

 cular distance of their common centre of gravity below the 

 upper surface of the fluid. 



Now, it is evident, that the same law would obtain if another point 

 were added to the system, and even if the number of points were to 

 become indefinite, or such that their aggregate or sum shall be essentially 

 equal to the area pressed, the law of induction would remain the 

 same ; consequently, if a denote the sum of the material points, or 

 particles of space in the surface on which the fluid presses ; then we 

 shall have 



p = Za. (1). 



This equation supposes, that the specific gravity of the fluid by 

 which the pressure is propagated, is represented by unity, which cir- 

 cumstance only holds in the case of water; therefore, let s denote the 

 specific gravity of any incompressible fluid whatever, and the general 

 form of the equation becomes 



p=$as. (2). 



