THE EQUATIONS OF MOTION. [CHAP. I 



and similarly for the remaining edges. Since the angles of the 

 parallelepiped differ infinitely little from right angles, the volume 

 is still given, to the first order in St, by the product of the three 

 edges, i.e. we have 



Dv .,, f, /du dv dw 





= \ 1 + (f + f + ^n t 



( \d% dy dz) 

 1 Z)v du dv dw 



Hence (1) becomes 



Dp (du dv dw\ 



-7^7 -r p -; -- h j -- r~j- I U .................. (o). 



Dt r \dx dy dz) 

 This is called the equation of continuity. 



rp, . du dv dw 



The expression -7- + -y- + -^- 



c?^ 1 c?y c?5 



which, as we have seen, measures the rate of increase of volume 

 of the fluid at the point (x, y, z\ is conveniently called the 

 expansion at that point. 



8. Another, and now more usual, method of obtaining the 

 above equation is, instead of following the motion of a fluid 

 element, to fix the attention on an element Sxby&z of space, and 

 to calculate the change produced in the included mass by the 

 flow across the boundary. If the centre of the element be at 

 (#, y, z), the amount of matter which per unit time enters 

 it across the i/^-face nearest the origin is 



and the amount which leaves it by the opposite face is 



The two faces together give a gain 



per unit time. Calculating in the same way the effect of the flow 



