Current Progress in the Slender Body Theory 



( FS) 



C = pV' 



33 



B'(L/2)B'(-L/2)K(L) + B'(L/2) B"(x) K(x-L/2)dx 



I 



/-L/2 

 •'-L/ 2 



B'(-L/2) I B"(x) K(x + L/2)dx 



-/ 2 



where 



L/ 2 rL/2 



•'- L/ 2 -'- 



B"(x) B"(x) K(x- x') dxdx' 



L/ 2 -^-L/ 2 



K(x) 



^f 



-^t. 



<D(m) 



o\ cos mx - 1 



dm 



^Jj-m^ |-n.3| cm + ajg)'* (cosmx- 1) 



dm 



(m+a)Q)'*(cosmx- 1) 



[^ J-oo J-ra^ ■^"■aj m'* y (m+ co^) 



dm . 



4 _ „2 ^2 



Since the above expression involves divergent integrals, the finite part of the 

 integral should be taken. 



COMMENTS ON SLENDER BODY THEORY 



E. V. Laitone 



Professor and Chairman 



University of California 



Berkeley, California 



It should be noted that the singularities noted in the integrals for the source 

 distribution can be always evaluated by using Hadamard's concept of the "Finite 

 Part" of the integral. This is a generalization of Cauchy's "Principal Value," 

 and can usually (but not always) be most simply determined by "Integration by 

 Parts." 



165 



