A Vortex Theory for the Maneuvering Ship 

 d) From par. 14, Eq. (17), we get: 



AL *^''35app ^f^-^c) " AL ^''"35 +^^g) - ^glirJ/lljJ (without planes) , 



AL ^'^35,pp+/^"g) = AL ^"35+^^g) - ^^[-ujl[-(j 



- ^ :f ^L a2Bg2B (%)/(lj) (with planes) , 



M5) 



(m+Mi) - bg = ^ (m+Mi) - b 



AL 



3PP AL 



2W 2W 



AL (M+/^i) - b^pp = — (M+Mi)- b 



1-0(0) - f(^ 



1-0(0) - f 



(without planes) 



+ A -;p Cj ^p+ a,pjf, (-pp] (v/itl-i planes) , 



A "Lg ^B ' ''2B' 2 \ u 



e) The first equation of Eq. (18) gives 



(Vj5 -AiCc) = Tf (^15-A^^g) + bgj 



AL ^15 ^""' AL 



f) Lastly, Eqs. (19), par. 14, yield 



u// \ u 



(6) 



2W 2W , , " \ U , 



AL ^^23pp+^>^2) = AL ^^2+MV2) + b ^^|-7^ (without planes), 



2W 2W 



AL ^^23pp+M>'2) = AL f^2+MV2) + b' —^ 



(ojL 

 ^T (^B%-c„ ja^B ^L/U ^^'*'^ planes), 



M7) 



b' = b' 

 app 



b^pp = b' 



, /<iiL) 



i-0'(O) - r 



1-0'(O) - f (^) 



(without planes) , 



T (^B-L -X) 'B+«2Bf2(T) 



(with planes) . 



891 



