the whole structure. If we let n be a unit normal vector out of the fluid, 

 then 



X^ = I p cos(n,x)dS 



Y^ = I p cos(n,y)dS 



Zj^ = I p COS (n, z) dS 



A- = 



i ~ j p[ycos(n,z)-zcos(n,y)]dS 



■ = p[zcos(n,x)-xcos(n,z)]dS 

 •'Si 



r. = I p [x cos (n, y) - y cos (n, x)] dS 



B- = 



[12a] 

 [12b] 

 [12c] 

 [I2d] 

 [I2e] 

 [I2f] 



.th 



The integrals are taken over the instantaneous surface of the i spar. 

 Here cos(n, x) is the cosine of the angle between n and the x-axis, etc, 

 We find readily that, to first order in small quantities, 



cos (n, x) = — cos X.' + v sin X.' + (3 a." 



cos (n, y) = — -y cos X-' - sin X-' — a a-' 



cos(n,z) = Pcos X-' - a sin X-' + a-" 



For abbreviation, we also define two sets of integrals 



.0 



Sn = f S(z/)(z.')^dz.' 



T^ = ( e^^i S(zp(z')^dz' 



[13a] 

 [13b] 



11 



