OF THE MOTIONS OF FLUIDS. 281 



dwz: --— d^ + --r— Md^ + -— - vdt + — — M;df ; 

 d^ d^ dj/ dz 



d'v , d'M ,. d'u ,^ . d't? j^ J 



du := dt+ -T- Md< + -r— rdf + -r— 2^d^ ; and 



df ax ay dz 



du? = - — d^ -\- -—udt + -— vdt + -7— M/d^ 

 d^ dx dy dz 



" The fluid being supposed to be divided into infinitely 

 small rectangular parallelepipeds, of which the sides are 

 parallel to the coordinates, we have, for the volume of the 

 element corresponding to ar, y, and z, [dxdi/Dz, using the 

 characteristic D with regard to the variations of space for 

 the same instant of time, while a and d are employed for 

 the successive changes only.] The density of the fluid 

 may be considered as constant throughout this space, and 

 may be called p, so that the mass will be fDxDj/Dz. We 

 may also designate by p the pressure, on each unit of the 

 surface, exerted by the fluid in contact with the different 

 faces of the parallelepiped, and which, according to the 

 fundamental property of fluids, is the same in all direc- 

 tions. The two quantities, f and p, as well as the veloci*- 

 ties u, V, Wj are unknown functions of x, y, z, and t; the 

 five quantities, u, v, w, f , and p, are required to be found 

 for the solution of the problem ; and when these have been 

 obtained, in terms of ar, y, z, and t, the state of the fluid will 

 be known for every instant, the velocity and direction of 

 the motion of each particle being determined, together 

 with the density of the fluid and the pressure exerted, 

 whether at the surface or within the substance of the fluid. 

 We must therefore proceed to seek for the equations ex- 

 pressing these five quantities. 



