On the general Differential Equations of Hydrodynamics. 437 



is deduced, by means of D'Alembert's principle, from the general 

 hydrostatical equation obtained as the solution of Prop. TIL, 

 just as questions relating to the motions of solids are solved as 

 statical questions by the intervention of the same principle. 

 The following is the analytical expression of this equation in its 

 most general form : 



It is here to be remarked that this equation, as well as the 

 hydrostatical one on which it depends, was investigated with refer- 

 ence to a single elementary particle. But as the particle might 

 be any one whatever of the mass of fluid considered, we may at 

 once assert, with respect to the hydrostatical equation, that it 

 applies to the whole of the mass. The same assertion cannot be 

 made respecting the hydrodynamical equation (1), unless there 

 be fulfilled certain conditions arising out of the distinctive cha- 

 racter of the motion of fluids, according to which the particles 

 move inter se } and continually change their relative positions. 

 In fact, that equation has no application unless such motion be 

 consistent with the principle of constancy of mass. This prin- 

 ciple requires the investigation of a general equation, which 

 shall express that each given element changes form and position 

 by reason of the motion in a manner consistent with its remain- 

 ing of the same mass in successive instants. The result of the 

 investigation, which answers Prop. V., is the equation 



dp d.pu ,d.pvd.pw_ 



dt + ~d^ + ~dy~ + ~dV-°' • ' * W 

 This, in case the fluid be incompressible, becomes 

 du dv dw __„ 

 dec dy dz 



4. Again, the movements of a fluid must be such as to satisfy 

 the geometrical condition that the directions of the motion in 

 each given element are normals to a continuous surface. It will 

 not perhaps be denied that, unless this condition be satisfied, 

 neither of the equations (1) and (2) has any application. But 

 the necessity of obtaining a general differential equation to express 

 the fulfilment of this condition has not been generally recognized. 

 I propose, therefore, before proceeding to the investigation of 

 such an equation, which, in fact, is the third general equation 

 mentioned above, to give some account of what has been done 

 with the other two, as this statement will serve to show the 

 necessity for the third. First, I remark that the two equations 

 have been applied to problems in which the motion is assumed 

 to be in directions tending to or from a fixed point or a fixed 



