

OF THE MOTIONS OF FLUIDS. 287 



quation of equilibrium, and it will become S/) = f J SxfP — 



)MQ-'^)^^'{R'-'^)} ■' o^ apposing 

 Pdx+Qhj-^-R^z to be an exact variation, and equal to 



f dt- ^ dt^ d^2 



364. CdiiOLLAiiY. Since the three varia- 

 tions are independent, their coefficients may- 

 be made to vanish separately, and the theorem 

 may be resolved into three distinct equations. 



S65. Theorem. The condition of the 

 continuity of the fluid is expressed by the 

 equation p^=:(f)i' (G) 



(f) being the initial value of the density f , 



and ^— ^'^ ^'^ ^^ _d'a: d'y d'z d'a* dy d'z d'a* d'y d'z 

 da db dc da dc db d6 *dc *da ' db da'dc 

 da d'?/ d'z d'a d'y d z , , • •, • i , r 



■^d-cTa'db-^c'lTb'd-a' *^ ^^^^^^1 ^^1"^S «f ^'3^' 



and 2: being expressed by a, i, and c, which 



are variable from particle to particle only. 



The coordinates, a, y, and z, are functions of the primi- 

 tive coordinates a, b, c, and of the time t: [it is evident, 

 for example, in the propagation of a wave, that the motion 

 of any particle, to which the ordinates a, y, and z belong", 

 depends entirely on the initial state of other ordinates of 

 the surface of the fluid, in combination with the time 

 elapsed from the beginning of the motion :] consequently, 

 [if the variations 3 be taken with respect to any one instant 

 of time,] 



