Ogilvie 



rX n = n^l + n^J + ngk (2- 72b) 



where 



r = xj + yj + zk. 



In particular, note that: 



n = n'k and n^ = znj - xn_. (2-72') 



2) <^i: This Is a normalized velocity potential. It satisfies: 



<}>ixx + 4>jyy + '^izz - ^ ^^ ^^^^ region; 



-^ = n: on S{x,y,z)=0; > (2-73) 



on ' ' 



I V^i I ~^ ^ ^^ infinity. 



3) v(x,y,z): This is a normalized fluid velocity, equal to the 

 fluid velocity at (x,y,z) when an incident stream flows 

 past the body, the stream having unit velocity, i , at 

 infinity. It can be represented as follows: 



v(x,y,z) = V[x - <^,(x,y,z)] . (2-74) 



4) m;: This quantity is related to the rate of change of 

 v(x,y,z) in the neighborhood of the body, as follows: 



m, i + mgj + mjk =m=-(n'V)v; (2- 75a) 



m4i +m5J +m6k= - (n • V)(r Xv). (2- 75b) 



In particular, note that: 



•z _ 

 m_ = ^ 



3 8n ^'zn' 



(2-75a') 



m^ = - 



8]E^(J' r Xv) = - -g^(zv, -XV3) 



= -|^[z(i-4,^)+x4>,J =-n3-H^(z<|>,^-x<f>,^). (2-75b') 



724 



