4 THE EQUATIONS OF MOTION. [CHAP. I. 



instant at all points of space occupied by the fluid; whilst for 

 particular values of x, y, z they give the history of \vhat goes on at 

 a particular place. 



Now let F be any function of a, y, z, t, and let us calculate the 

 rate at which F varies for a moving particle. This we shall denote 



T^Ti 1 7\ 



by -^- , the symbol ~- being used to express a differentiation 



following the motion of the fluid. At the time t + dt the particle 

 which at the time t was in the position (x, y, z) is in the position 

 (x + udt, y + vdt, z + wdt}, and therefore the corresponding value 

 of F is 



i- dF J* dF * dF 7, dF 7, 



F+ -y- dt + -s- udt + -7- vdt + ~r wdt. 

 dt dx dy dz 



Since the new value of F for the moving particle is also ex- 



1F 



dt 



dF 

 pressed by F + -~-- dt, we have 



dF dF dF dF dF 



-57 = -77 +U-J- + V-J- + w-j- (1). 



dt dt dx dy dz 



6. Let p be the pressure, p the density, X, Y, Z the compon 

 ents of the external impressed forces per unit mass, at the point 

 (x, y, z) at the time t. Let us take a rectangular element having 

 its centre at (x, y, z}, and its edges dx, dy, dz parallel to the co 

 ordinate axes. The rate at which the ay-component of the momen 

 tum of this element is increasing is pdxdydz -; and this must 



be equal to the ^-component of the forces acting on the element. Of 

 these the external impressed forces give pdxdydzX. The pres 

 sure on the i/^-face which is nearest the origin will be ultimately 



[P &quot;2 f dx ) dydz, that on the opposite face f p + J -~ dx\dydz. 



The difference of these gives a resultant J dxdydz in the di 

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

 perpendicular to x. We have then 



p dx dy dz x = p dx dy dz X - f~ dx dy dz. 



