40 Colorado College Studies. 



in the same directions; and finally let j^ represent the pres- 

 sure at the point {x, y, z) and in the mass of the element. 

 From the conditions of the problem 



"=/ (•^' Ih ^» ') 

 and it follows immediately that 



^n (hi , (In , dn . dn , 



— = \-u \-v~--\-w — =jx . 



St dt dx dy dz 



This last is an expression for the effective or expressed force 



in the direction of the x axis. In a similar notation — , or/„ 



dt 



and — , or fz represent the effective forces in the direction 



of the y and z axes respectively. 



The total impressed forces are mX, mY, and mZ; the total 

 effective forces 7»/,., fiify, mfz; the pressures on the faces 

 AC, AE, and AF are p dy dz, p dy dx, and p dx dz respec- 

 tively: on the faces opposite these the pressures are p dy dz-\- 



— dx dy dz, p dy dx-\- — dz dy dx, and _p dx dz + ~- dy dx dz. 

 dx dz dy 



The excesses of pressure on the opposite faces are: 



— dx dy dz, — dx dy dz, and — dx dy dz. 

 dx dz dy 



Since the hypothesis excludes the possibility of any loss 

 through friction, we have now considered all the forces in- 

 volved. Equating separately the forces acting in the direc- 

 tion of each axis by D'Alembert's principle, we obtain the 

 following equations: 



mX—mfx — — dx dy dz=Q 

 dx 



mY—mfy dx dy dz=0 



dy 



mZ—mfz — dx dy dz=0 



dz 



Substituting for m its value /' dx dy dz in which f repre- 

 sents the density of the fluid, and then dividing by dx dy dz, 

 the resulting equations are: 



