EQUATIONS OF MOTION 



59 



moves from point to point of its path, since this now also includes the 



ariation of x, y, z with t, its components will now be represented by -y- 



d t 



c., and are the components of the true acceleration of the particle. 



hese are not necessarily zero for steady motion. 



If then a particle moves from a point (x, y, z) to a second point 

 + 8 a? ; y + 8 # ; z -\- b z) in time 8 t, its change of velocity 8 u is given 



<N 9 u ^ 



O tt = = . O 



. o u ~ 



+ ,5 



r a 



+ ^.s 



nd in the limit 



d u 9 u d x 



d t 9 x ' d t 

 But from the definitions of u, v, w, we have 

 d x 

 dt 



9 u d y 9 u d z 8 u. 

 dy' dt * d z'dt dt 



(1) 



u = -= 



w = 





milarly 

 id 



d u _ d u 



~7 I - ^' O "l 



M; 



u u u 



l" ^' O l" O I 



w . d w 



(2) 

 (3) 



(4) 



Next consider an elementary parallelepiped in the fluid bounded by 



dges 8 x, 8 y, 8 z (Fig. 30). For 



ntinuity of motion the difference 

 itween the amounts of fluid which 

 )w into and out of its faces during 

 me 8 t must be equal to the in- 

 ease in the mass which they 

 iclose. Expressing this analyti- 

 lly, we get the equation of 

 ntinuity. 



Now the mass of fluid entering 

 ross the face B in time 8 t = 

 \i 8 y . 8 z . 8 *. 

 And the mass of fluid leaving 



9 

 ross the face C D in time Bt = puBy.Bz. B t + ^ ~ (p u) 



.'. Gain across these faces == ~ | (p u) & x . S y . 8 



FIG. 30. 



