5* PRESSURE AND ENERGY 361 



and each carrying with it the momenta mu, mv, mw, the total 

 momenta carried across in the backward direction are 



ro f 10 f 



dy dz Nm du\ dv\ dw u*F(u t v, w), 



J _oo J CO ^ CO 



ro r 00 r 



dy dz Nm du dv dio uvF(u, v,w), 



J oo J CQ J oo 



ro r 00 r 00 



dy dz Nm du\ dv\ dw uwF(u, v, 10). 



J- J -co J -co 



6*. Components of Pressure 



The second half of the medium loses these latter momenta 

 and gains the former. The resulting increase of its momentum is, 

 therefore, given by the difference of these expressions, so that 



/OO fOO /-CO 



dy dz X x = dy dz Nm du dv dw u 2 F(u, v, w) , 



J _oo J -co J -co 



/OO /-00 pCO 



dy dz Y x = dy dz Nm du dv dw uvF(u, v, iv), 



J OO J CO J CO 



pOO /-CO pOO 



dy dz Z x = dy dz Nm du dv dw uwF(u, v, iv) 



J 00 J CO J 00 



are the components of the momenta which, during the unit of 

 time, pass over the surface dy dz from the first half of the medium 

 to the second, or, more briefly, are the components of the force 

 exerted on dy dz by the first half towards the second. 



Just as we have here found the force-components which act 

 on a surface perpendicular to the #-axis, and are denoted by the 

 suffix x, we may obtain the analogous magnitudes for the other 

 two axial directions. We thus get, with the corresponding 

 notation, the following values for the forces exerted per unit 

 area, that is, for the pressures, 



/CO pOO /-CO 



X x =Nm) duj dvj 



/CO |>CO fCO 



Y v = Nm du dv] dw v 2 F(u, v, w), 



J 00 J 00 > CO 



/CO pCO /-CO 



Z z = Nm\ du\ dv\ dw w 2 F(u, v, w). 



J _co J co J _oo 

 rco /-co />oo 



Y ' z = Z y = Nm\ du\ dv\ dw vivF(u, v, w). 



J _co J_oo J oo 



rfCO /-CO 



du dv dw wuF(u, v, w), 

 ^CO * CO J 00 



/CO /CO pCO 



Xy Y x = Nm du dv\ dw uvF(u, v, w). 



J -CO J -CO J -00 ' 



