196 CELESTIAL MECHANICS. I. iv. 17. 



y-^y -\- -vT ^^]* Now J5 being a possible and consistent quan- 

 tity, its variations, and consequently its fluxion, must be 



exact (315): we have therefore (314), £(f£):=4Mi ; 



ay dx- 



d'(pP)_d'(pJ2) d'(fe)_d'(fi?) ,, r • ^v PN 



— f— -=--^ — ; — ^-^=-^-—; consequently r, since d(pjr) 

 dz da: dz dy 



=:fd'P + Pd'f , ..., we have, by combining the three last equa- 

 tions, multiplied by P, Q, and 12, 0= P (o^+Q^ - p 



\^ dz dz 



(p -1 — + P ~— p — ;^ — --^\ : and since the terms con- 

 ^ dy dy dx dx/ 



taining d'p obviously destroy each other, we obtain, from 



those which are multiplied by f, the equation] 



dz dz dy dy dx djr 



And this equation expresses the relation between the 

 forces P, Q, and R, which is required in order that the 

 equilibrium may be possible. 



If the surface of the fluid, or any part of the surface, is 

 at liberty, the value of j;? must be evanescent at that point, 

 since there is no pressure that could be measured by p; 

 we have therefore for the direction of the surface 3|]pzzO, 

 the variations ^x, ^y, ^z, being so related as to belong to it. 

 The independent forces must therefore balance each other 

 with respect to any motion in the direction of the surface, 

 and O—P^x + Q^y + Rh: but this c^n only happen when 

 the result of these forces is perpendicular to the surface, 

 the general equation SiS5i? + ** R" hzzO{c) (252) becoming 

 here P^x + Q^y + R^z + '' R' ^r-O, and P^x + Q^y + Rh 

 =1— " i^" J"r, indicating a result in the direction of r, the 

 perpendicular to the surface. 



