Ogilvie 



f ik ' k = 1, 2, 3 , 



hk(x) = < (A7) 



I 1,^.3 XX, k = 4, 5, 6 , 



^ikCt) . p J^ -^^ ^_ - V 3^ + Vcp„ • Vj x,,(x,t) dS 



y f 30 (X) BCP^(X) / 3 3 \ 



^tJ^ -i^ 3;rr(Br-^3^j^ik(x,t)d^, (A8) 



■^ik^t) = P J^ —3^ (bT-^B^^ W, .Vjy,,(x,t) dS 



O 



^tJ^^S^ Sr(3?-^ls)'<-^w<'£.t)d-t- (A9) 



It may be noted that the quantities (Cj^ - c j^) arise from the transformation 

 of force and moment components from the unsteady to the steady reference 

 frames. 



There is some ambiguity in choosing the best representation of the convolu- 

 tion integral terms. There would be no basic difficulty in carrying along both 

 sums, but it would lead to much extra writing later on, and so we choose to com- 

 bine the two sums of convolution integrals into a single sum. We can partially 

 integrate the sum containing the L.,^'s: 



t t t 



CD - 00 



and then this sum has the same appearance as the other one, since the integrated 

 part vanishes. However, for reasons of convenience later, we choose now to 

 handle these integrals in the opposite way: We assume that a partial integration 

 can be performed on the sums containing the Mjj^'s and that the integrated terms 

 will vanish, so that we can write the last two sums in Eq. (Al) as follows: 



6 t 6 t 6 t 



E J ^^(r) L.^(t-r)dT + E r ai^(r) M.,^(t-T)dT = E f <^j^(t) K. ^( t - r)dr . 



k=l-oo k=l-^-oo k=l-co 



(AlO) 

 A further discussion of this point appears in Appendix B. 



74 



