11 



— 6. = - r„ X t* 



dt « 



Suminarizing, when we combine Equations [10], [13], [15], and [17], we have 





i(q • q)n - (q • n)q 



da 



J p[r(q.n 



) + ^n 



da 



-b 



(r X <i» • n)q - (r X w • q)n + O (n x w) 



did 



da 



[18] 



r X ^ - M (r • tc) ) + r ( 

 .^ dt ^ ^ 



ta • ot)\ 



pY- 



The first term in the above expression would give the force if the body were not rotat- 

 ing and the undisturbed stream were steady, i.e., the "Lagally force." The second term is 

 due to the change of the flow with time, and the last two terms arise when the body is in rota- 

 tion. Since these various components will be discussed separately, we call them F^, F,, F,, 

 respectively. 



■=l4i 



(q . q)n - (q • n)q 



:i9a] 



'^^-ij/h-"^^'^" 



da 



[19b] 



F3 = - j p [(r X 61 • n)q - (r X tj • q)n + $ (n X «j )] c?(T 

 - r„ X -^ - €u (r . w ) + r (fc> • a) 



\_ B dt S 6 



[19c] 



If the origin of the system of axes is taken to coincide with the centroid of the body, 

 the last term of the expression for F, vanishes. 



The above forces are defined in terms of integrations over the control surface S'. Since 

 S' has not been specified, it is evident that the forces are independent of the particular choice 

 of S', as long as it satisfies the conditions necessary for the integrations to be carried out. 



I THE "LAGALLY FORCE", Fj 



Initially, we suppose the singularities generating the body to be discrete, isolated, and 

 fixed with respect to the body. Their locations are designated by the set of position vectors 

 r .. For the control surface, we select a set of spheres S,- with their respective centers at the 



