4-7] EQUATION OF CONTINUITY. 5 



that on the opposite face 



(p + ^dp/dx . Sx) SySz. 



The difference of these gives a resultant dp/dx. SxtySz in the 

 direction of ^-positive. The pressures on the remaining faces are 

 perpendicular to x. We have then 



pxyz^- = pSxtySz X BxBySz. 



Substituting the value of Du/Dt from (1), and writing down 

 the symmetrical equations, we have 



(2). 



7. To these dynamical equations we must join, in the first 

 place, a certain kinematical relation between u, v, w, p, obtained 

 as follows. 



If v be the volume of a moving element, we have, on account 

 of the constancy of mass, 





To calculate the value of 1/v.Dv/Dt, let the element in question 

 be that which at time t fills the rectangular space &pfy* having 

 one corner P at (x, y, z), and the edges PL, PM, PN (say) parallel 

 to the co-ordinate axes. At time t + &t the same element will 

 form an oblique parallelepiped, and since the velocities of the 

 particle L relative to the particle P are du/dx . &, dv/dx . Sx, 

 dw/dx.Bx, the projections of the edge PL become, after the 

 time &, 



/-i du *.\ * dv ~ 5. dw ~ s 

 1 -h -5- dn &r, -j-St. So?, 3- Bt . &r, 



V dx J dx dx 



respectively. To the first order in &t, the length of this edge is 

 now 



, , du 2.\ * 

 1 + -j- o&amp;lt; I ox, 

 dx J 



