and 



q = - VA^ + v„ • r^) 



" m m o m 



Hence q satisfies a velocity potential which we designate by (f)^. It is related to <!> by 

 (t> (x , y , z , t) = <b(x -^ x„, y + y„, s„ + 2„, i) + v„ • r [3] 



The pressure at any point, not considering the gravity field and an additive function of 

 time, is 



p = -_p.q.q + p-=--p(q^.v„).(q^.V,).p_ [4] 



From Equation [3] 



— = V7 $ . ^ .r - v„ • 



dt "• "" dt dt dt "" ° dt 



But 



dr dt 

 m c 



dt dt 



so 



d^ dy^ 



i*=q .V +.l_!i o.r +v . v„ [51 



~ ^m o ^, , m o o 



dt dt dt 



Substituting in Equation [4] and collecting terms, 



p = _i.pq .q +p 2L_p_^.r^ [6] 



^ 2^ " "" ^ dt ^ dt " 



in which a term containing v^ • v^ has been dropped. This is permissible, since v^ is a func- 

 tion only of time, and the net force or m.oment due to a constant pressure acting on a closed 

 surface is zero. 



If the velocity field q were to be considered absolute rather than relative, and the 

 pressure were calculated accordingly, we should have 



(9$ 

 m 



dt 



SO we can write 



1 



^ = ^'"-''^ 



[7] 



By means of this relation, the flow relative to the m.oving axes can be considered as if it were 

 the actual flow, and the forces and m.oments so obtained can be converted to the true values. 

 Thus, for the force exerted on a given surface S, 



