slender body. At this point it is not profitable to attempt further to justify the linearization 

 but rather to throw the burden of the justification upon a comparison that will finally be made 

 between exact and linearized theory results for particular cases. 

 The linearized boundary conditions are 



dVo , . _ v(x.O) 



dx 



2u(x,0) 



c < X <0 

 < X < I 



[la] 

 [2a] 



Finally, then, the linearized problem may be stated: To find a harmonic function 

 :f>{x,y), symmetric with respect to the aj-axis, whose gradient in the limit vanishes everywhere 

 on a circle of sufficiently large radius about the origin, which satisfies the mixed boundary 

 conditions illustrated schematically in Figure 2, and which satisfies the additional condition 

 that 



I" ^ ix,0)dx = 

 The last condition assures that the body be closed, since 



dx U^ dy 



The mathematical meaningness of the problem is not known a priori, but becomes clear as the 

 solution is constructed. 



3d 3cS dVo , 



-x^(x,o) = o^^ -r-u,o)^u.-^{x)- 



oy ' oy ' c dx 



3rf ctU, 



ox ' 2 



3(Z^ +°° 



3y 



x = 



x=l 



Figure 2 - The Linearized Boundary Conditions 



A distribution of sources of strength m(x) along the a;-axis for -c < a; < Z produces a 

 harmonic function with the proper symmetry. 



