Ship Waves. 7 
5. The system we have just considered may be supposed to correspond to 
a ship with bluff bow and stern. We may examine the effect of pomting the 
ends by the following distribution ; the moment per unit area 
=M, for—a<a<a 
=M(l—2z)/(l—a), fora<a<l 
=M(l+2)/(l—a), for—l<a2<—a, (20) 
where M is a constant. The moment is zerbd outside the range specified in 
(20). 
If we replace M in (20) by ub/2zx, in accordance with (6), we see that if 6/1 
is small the corresponding form of ship is that examined by Wigley in the paper 
already quoted. Wigley has worked out the surface elevation along the line 
y =0 from Michell’s formule, giving suitable interpretations to the indeter- 
minate integrals involved in those formule. Here we shall use the general 
form (5). We may take the distribution to extend right up to the free surface, 
as it appears that the resulting expressions are finite and continuous through- 
out. 
From (5) and (20), after carrying out the integration with respect to f and h, 
the surface elevation for y = 0 is given by 
eqs esa | sect 6 d0 [ eC Aiea eer 
mu (l — a) Jo o K (kK — ky sec? 0 + ap sec 8) 
where 
N = as (+a) cos 0 __ et (c+l) cos @ __ Bix (x—l) cos 0 aL ei (%—a@) cos a (22) 
We notice from the form of N that the singularity at « =0 in the integral 
with respect to « is only apparent. On the other hand, the integral as it 
stands cannot be separated directly into four parts associated with the points 
-+ta, --l respectively ; this may, however, be effected by a slight alteration 
which does not affect the final result for the complete system. 
If we write 
4M 7/2 5 io) ih ekd cos 8 
———— 648 |. ————_________ 
$(@) ail et o K (kK — kg sec? 6 + ty sec 0) < 
4M 
= ———_ ff 
ao (23) 
then we have 
C=S(@—1) —C(e@—a) -—Cw+a)+C(e+)). (24) 
The integrals in (23) may be transformed in the usual way to separate out the 
353 
