

10 THE EQUATIONS OF MOTION. [CHAP. I 



Equation of Energy. 



11. In most cases which we shall have occasion to consider 

 the extraneous forces have a potential ; viz. we have 



<m <m <m 



A = -- = , i = -- 5. Z/ = -- ^ ............ (1). 



dx dy dz 



The physical meaning of fl is that it denotes the potential energy, 

 per unit mass, at the point (x, y, z), in respect of forces acting at 

 a distance. It will be sufficient for the present to consider the 

 case where the field of extraneous force is constant with respect to 

 the time, i.e. dljdt = 0. If we now multiply the equations (2) of 

 Art. 6 by u, v, w, in order, and add, we obtain a result which 

 may be written 



DH / dp dp dp 



- = - u+v + w 



If we multiply this by SxSy&z, and integrate over any region, we 

 find, since DjDt . (pSxSySz) = 0, 



where 



T= I////? <y + tf + O dxdydz, V = fffttpdxdydx . . .(3), 



i.e. T and V denote the kinetic energy, and the potential energy 

 in relation to the field of extraneous force, of the fluid which at the 

 moment occupies the region in question. The triple integral on 

 the right-hand side of (2) may be transformed by a process which 

 will often recur in our subject. Thus, by a partial integration, 



I / 1 u d dxdydz = 1 1 [pu] dydz Illp-r- dxdydz, 



where [pu] is used to indicate that the values of pu at the points 

 where the boundary of the region is met by a line parallel to x are 

 to be taken, with proper signs. If I, m, n be the direction-cosines 

 of the inwardly directed normal to any element SS of this 

 boundary, we have 8ySz=lSS, the signs alternating at the 

 successive intersections referred to. We thus find that 



ff[pu]dydz=-fJpuldS, 



