of the Motion of Fluids, Sfc. 387 



such a manner that it may be considered solid, (f> has no ex- 

 istence, and the pressure is determined by 



p =f{Xdx + Ydy + Zdz), 



the forces, X, Y, Z, including those aiising from rotation. 



We have then for the solution of any proposed problem, the 

 equations, 



^ dt na(x^ "^ df'^ dzy 



i^), 



<p =f{t) + -=_£ifL__ =fit) + 1^ (B) 



d4 ^ - F(,t) F{t) 



dr (x-af + (y-/3)' + (z-7)»- 



^^ 



(Q. 



Fit) 

 The expression V-^ for the velocity, is to be taken with 



respect only to a portion of the fluid, for vj'hich a, fi, y, and the 

 form of F remain the same, while r varies. This portion will 

 in general be elementary ; and the expression above will have 

 to the general expression for the velocity, which must be a 

 function of x, y, z, and t, a relation analogous to that between 

 the two expressions for the .same variable, derived from the 

 general and the particular solutions of a common differential 

 equation. The complete integral of (2), supposing it obtained, 

 would shew, as it must contain arbitrary functions, that no 

 necessary connexion exists between the velocity of one elemen- 

 tary portion and that of another contiguous to it, but such only 

 as we choose to impose by vessels, pipes, or other means. 

 Hence the form and value of F{t) may change at a given 

 instant from one portion to another; they may also change in the 

 .same portion from one instant to another: and a, /3, 7, may 

 change in like manner. The equations {A), (B), (C), are con- 



