and, in a similar fashion, 



-H 

 + X"M^e^(^+^l)jo(kr)dk\dzi [14] 



+ TTu^K^ f [R(zi)]^(zi-zc) e^<^"^^l^f-|-J (Kr)ldz^ 

 J_H '- -' 



+ ^°'k^e^(-+-l)j^(kr)dk}dzi [15] 



+ ttcK; r° R{zi)^ eK{z+Zi) j^(Kr)dZj^ 

 J_H ^^1 



^^ ^ _ i <^ r e^'^l <U KR + ^) R cos o.t 



[16] 



+ r2 sinojt — + ^KR"* cosGot-^ W[r2 + (z - z, )^] 

 9x 9x2j V ^ 



^ r"k+K k(z + zi) ^ ,, ,j,l, 

 ^ fo^^ Jo(kr)dk|dzi 



-.cK f' e^(^-^2zi)|(iKR + |^)Rsin.t 



- r2 coswt •^+ ^ KR"* sina3t-^l jQ(Kr)dzj 



where 4- denotes the Cauchy principal value. From the Appendix we see 

 that the potentials [13] to [16] satisfy the boundary conditions [8] to [H], 

 respectively, with a maximum fractional error of order R. Unfortunate- 

 ly, this error is not so small as in the classical slender-body theory for 

 an infinite fluid, where the error is of order R log R; for this reason 

 the present theory may not hold for as wide a range of slenderness as in 



7 



