A Vortex Theory for the Maneuvering Ship 



^.0 = i^ALU^f-: (U),, - "^'^ (t),, 



511 = ipALU^ [a^ + a^ (^) 



n -- 2 PALU^ 





, /Lr 



^ (1) 



i iu/., ^ '^1 vuy., 



The effect of the delayed circulation gives a system of forces 



X = X , - AX, 



where 



AX = I pAU^J ^ 



AY - - ^ pAU^ ^aj 



+ b 



AZ = - 2 pAU^ i a 



t ' 



fokCt') 0(0) + [ foi^(T') ^(t'-T')dT' 



E fokCt') 0l(O) + [ foi^(T') 0,(t'-T')dT' 



t ' 



(U),, ^1^°^ ' \ (U) , ^l(t'--')dr' 

 ^ '^ 



L t '^ -" 



t ' 



U) , ^^°^ +1 (o) , <^t'-r')dT' 



jV^)^,^(0)./'(^)^^^(t'-.')d.' 



(2) 



(3) 



(Cont.) 



A£ = i- p ALU' ^ ma. 



+ m'b , 



(g) 0,(0) . f (^) ^ 0,(t'-r')dx' 

 "^ 



if),, '^l^O^ ^1 (if)^, 0l(t'-r')dr' 



871 



