28 Prof. Challis on the Principles of Hydrodynamics. 



triangular section, « and /8 including the right angle, and let 

 Picth, Pz^h, pgjh be the respective pressures on the three rect- 

 angular faces. The element being indefinitely small, the pres- 

 sure may be assumed to be uniform throughout each face. Sup- 

 pose the impressed accelerative forces, resolved along the sides « 

 and /S in the directions towards the right angle, to be/i and/g. 



The impressed moving forces in the same directions are ^/,pa/3A 



1 . 



and ;r fcfiu^h, p being the density of the element. These must 



be in equilibrium with the pressures on the rectangular faces 

 resolved in the opposite directions. 



The pressure resolved in the direction of the side «, and tend- 

 ing from the right angle, is 



P^^h-2o.fihx -, ov ip2-ps)^h. 



The pressure resolved in the direction of the side /3, and tend- 

 ing from the right angle, is 



Hence 



and 



Picch-psjhx -, or {Pi-P3)cch. 

 ilh -P-6)^h = ^fpu^h, or p^ -p^ = -^ ; 

 {p^-ps)uh=-^f^px^h, orp,-;j3=-^. 



Consequently, as a and /3 are indefinitely small, the right- 

 hand sides of these equations are indefinitely small, unless /, and 

 /g be indefinitely great, which is assumed not to be the case. 

 Hence j0j=j02=/'3. By supposing the position of 7 to be fixed, 

 and those of « and j3 to vaiy in any manner so as to remain 

 pei-pendicular to each other, it may be inferred from the foregoing 

 reasoning that the pressures in all directions from the element 

 in a given plane are the same. Supposing another plane to pass 

 through the element, it may be similarly shown that the pres- 

 sures in all directions in this plane from the given element ai"e 

 the same, and consequently the same as the pi'cssures in the 

 other plane, because the two planes have two directions in com- 

 mon. And as the second plane may have any position relatively 

 to the first, it follows that the pressures are the same in all di- 

 rections whatever from a given element, or from a given point. 

 This is the law of pressure which it was required to investigate. 



It has been justly remarked by the author of the Treatise on 

 Tides and Waves in the Encyclopcedia Metropolitana, that the 



