4-7] EQUATION OF CONTINUITY. 5 



that on the opposite face 



(p + \dp\dnc . Sx) SySz. 



The difference of these gives a resultant dpfdx. &x SySz in the 

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

 perpendicular to as. We have then 



p&x&ySz ^ = pSxSySz X - & SxSySz. 

 jJv djX 



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, 





Dt 

 1 Dv 



To calculate the value of l/v.Dv/D, let the element in question 

 be that which at time t fills the rectangular space SxSySz having 

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

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

 form an oblique parallelepiped, and since the velocities of the 

 particle L relative to the particle P are du/dx.fa, dvfdx.Sx, 

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

 time $t, 



du \ dv dw 



dx ) ' dx dx 



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

 now 



-fi . ^ 

 Tda> 



