10 THE EQUATIONS OF MOTION. [CHAP. I. 



parallel to a; is found as in Art. 6, to be 



( Y^\d d d (16&quot;) 



Hence, (14) is now equal to (15) and (16) combined, which 

 gives 



d . pu d. pu? d . puv d . puw _ Y G 



dp 



dx ......... 



Performing the differentiations, and simplifying by means of the 

 equation of continuity, we are led again to the first of equations (2), 

 and in like manner the second arid third equations may be ob 

 tained. 



13. Another interesting application of the method of Art. 

 12 is to make Q (^(f+ V+E)p, the energy per unit mass. 

 Here q denotes the resultant velocity tj(u 2 + v 2 + w 2 ), V the 

 potential energy per unit mass with reference to the external im 



pressed forces (viz. we have X= -7 , &c. J , and E the intrinsic 



energy. In a liquid we have E=0. If the system of external 

 forces do not change with the time the alteration in the energy 

 contained within the space dxdydz is due to the flow of matter 

 carrying its energy with it, and to the work done on the con 

 tained matter by the pressure of the surrounding fluid. The 

 total rate at which this pressure works is 

 d.pu d.pv d. 



The verification of the formula obtained by equating (14) to the 

 sum of (15) and (18) is left as an exercise for the student. 



14. To obtain by the same method a proof of the surface- 

 condition (11) of Art. 10, let in Fig. 1 (Art. 3) P denote a point 

 of the fluid infinitely close to the surface F=0 , and let A, B } G 

 be the points in which this surface is met by three straight lines 

 drawn through P parallel to the axes of co-ordinates. Then if 

 PA, PB t PC = a, ft, 7 respectively, we have 



where F denotes the value of the function F at P (x, y, z). The 

 rate of flow of matter into the space included between the three 



