4 THE EQUATIONS OF MOTION. [CHAP. I 



any two neighbouring particles P, P will always be infinitely 

 small, so that the line PP' will always remain of the same order 

 of magnitude. It follows that if we imagine a small closed surface 

 to be drawn, surrounding P, and suppose it to move with the 

 fluid, it will always enclose the same matter. And any surface 

 whatever, which moves with the fluid, completely and permanently 

 separates the matter on the two sides of it. 



5. The values of u, v, w for successive values of t give as it 

 were a series of pictures of consecutive stages of the motion, in 

 which however there is no immediate means of tracing the 

 identity of any one particle. 



To calculate the rate at which any function F (x, y, z, t) varies 

 for a moving particle, we remark that at the time t + Bt the 

 particle which was originally in the position (x, y, z) is in the 

 position (x + uBt, y + vBt, z -f wBt), so that the corresponding value 

 of ^is 



F (x + u$t, y + vBt, z + wBt, t -f Bt) 



St St+wSt t 



dx ay dz at 



If, after Stokes, we introduce the symbol D/Dt to denote a 

 differentiation following the motion of the fluid, the new value 

 of F is also expressed by F + DF/Dt . Bt, whence 

 DF dF dF dF dF 



-j=rr = -jI + u ^- +V-J-+W-T- ............... W- 



Dt dt dx dy dz 



6. To form the dynamical equations, let p be the pressure, p 

 the density, X, Y, Z the components of the extraneous forces 

 per unit mass, at the point (x, y, z) at the time t. Let us 

 take an element having its centre at (x, y, z), and its edges Bx, 

 by, Bz parallel to the rectangular co-ordinate axes. The rate at 

 which the ^-component of the momentum of this element is 

 increasing is pBxByBz Du/Dt; and this must be equal to the 

 ^-component of the forces acting on the element. Of these the 

 extraneous forces give pBxByBzX. The pressure on the ^-face 

 which is nearest the origin will be ultimately 



(p ^dp/dx . Bx) ByBz*, 



* It is easily seen, by Taylor's theorem, that the mean pressure over any face 

 of the element 5x5y8z may be taken to be equal to the pressure at the centre of 

 that face. 



