Newman and Tuck 



"J. 'J_ 'J 



im(x-x) ^ ' 



1 yjm ^x^ - (m+oj. )'* 



dm 



where 



$(m) 



( m + oJq ) • 



.1 (m+ajg)' 



7x2 m2 - (m + aj^)4 



when x|m| > (m+oj)- 



( m + Wq ) 



V^m+aJg)'* -x^: 



cosh 



_i (m+ajg) 



when x|m| < (m+ai. )■ 



Making use of the above, the hydrodynamic forces can be expressed by con- 

 vergent forms. The component of the force in heave for instance, is given by 

 the following form: 



,(FS) ^ pU^ 



''' ~ 2T.2 





dm 



L I 



3I 2 (m+cjQ)*dm 



i^V^m^ x^ - (m+ a; )4 





( m + w. ) ^ dm 



,/(m+c<jQ)'' -m^x- 



where 



P. = I ^^ e^'-'^dx 



L/2 



dx 



This formula resembles Michell's integral for the wave resistance in uniform 

 motion. Hanaoka has given a similar formula for the hydrodynamic forces and 

 moments of an oscillating thin ship. The discussor wishes to propose that the 

 above formula will be called Hanaoka 's integral. There is another type of the 

 expression, which is given by repeated integrals of a kernel fvinction and has 

 some resemblance to Vosser's formula for the wave resistance of a slender 

 ship. 



164 



