Brard 



it's , ^0 '^L I 4. : 7T 



\J6 + — — — COS cijt + — 



for w/u, and 



L lir) '^^ 



s oJt 



for Lw/U^ in formulae of par. 8. 



We obtain, for instance: (given by the first formula 8) 



Z = Z + ipAU^ j^ (M+M3)(^l cosa;t- 



(which comes also from the contribution of the quasi -steady motion); 



- COs(a,t +-^| ^ 



Z = Z + ^ pAV^a \ 0(0) ^ cos (wt + - I + 



6jl r' 



u J 



COS (^ t' + ^)0(t'-r')dr'f^ 



(which gives the effect on the body itself of the delayed circulation around the 

 body); 



^ . 1 



6jL 



Z = Z + - pAl}^ 1 ^ T ^Lr^ IB U cos ( a)t + 2 



^T"%^.Bf / c„s(^.'.f)i„(f-.-)d.' 



(which gives the effect on the diving planes and fins of the delayed circulation 

 around the body). 



Because the harmonic motion is assumed to be perfectly established the 

 interval (0, t ') is infinitely wide, and consequently, the lower limits in the 

 integrals above must be taken equal to -<». But we have 



I cos (^ r' 



t '- r') dr 



a 



-I 



"U (t -r ) + - 



^t' ) dr' 



:os.txg(^) + cos (-t + f)f(f) , (9) 



where 



are the cosine and sine Fourier transforms of the derivative <P(t') . 



886 



