42 



(x-x )(x -x) 



:(*> *)-'(*) (t-xj(x-t) ti° 6 l 



The length -diameter ratio X of the spheroid is given by 



, (t-x )(x x -t) 

 X " f(t) 



:io7] 



whence the longitudinal virtual mass coefficient k (t) can be obtained from 

 [26]. 



Since U(x) = (1 + k ) cos y(x) is an exact solution of [94] for the 

 prolate spheroid, we have 



J\'(x, tjdx^^^y [108] 



We now obtain, from [98], [100], and [108] 



E,(t) = 1 - ^(^[klx, t) - k'(x, t)]dx - 1+k { 1 t) [109] 



Also [103] may be written in the form 



E n+1 (t) = E n (t) -jpkfx, t)[E n (x) - E n (t)]dx - E n {t){X(x, t)dx 



x x 



But from [98] and [100], 



ri 1-E (t) 



k(x, t)dx = 2 ^ k 



Jx Q 1 



Hence we obtain 



E.(t)+k, i ri r 1 



E n + l (t) " "W" E n (t) "tJ k(x ' t] K U) ' E n (t, ] dx [110] 



1 



ILLUSTRATIVE EXAMPLE 



The present method will now be applied to the same profile [78] as 

 before. By way of contrast with the semi -graphical procedures previously used, 

 a completely arithmetical procedure will be employed. 



