[8] 

 continued 

 + A e ^ (a-' + I ka- ) sin (kag cos S- - wt)] 



+ a. [xQ+(3t,-YaQsin9-+Ae cos(kaQCOs9^-ojt)] — 



9xj 



9 a 



+ a. [yq- Q-C + \aQ COS 9-] — 



+ i af [kAe^^ sin (kag cos 9- - ojt)] -^ jje^^^i^^ ^ Jq (kR. )| d^ 



In accordance with the assunnptions of slender body theory, we eval- 

 uate these terms as R^ -* and identify the values so obtained with the 

 potential on the body surface R^ = a^ . Because a-/aQ « 1, the value of 



$ 



to lowest order on the i spar depends only on the first term above and 

 one term in the first sum. By the same approximation procedure, we find 

 that all terms in the second sum contribute amounts of higher order in 

 terms of a^. Later we shall reconsider the second sum when we calculate 

 dannping forces. 



FIRST-ORDER FORCES AND MOMENTS 



The pressure is obtained from Bernoulli's equation in linearized 

 form 



9$ 

 P = - P -^ - Pgz [9] 



Thus it will be necessary to evaluate $, on each spar and to integrate, 

 in an appropriate way, the result over all spars to find the forces and 

 moments . 



fh 



Using slender body approximations again, we find that on the i 



spar 



