Ogtlvte 
is \d| [ : nid 
These three transforms can now be combined to yield the result stat- 
ed above, thatis, 
9o 
a ( - 0) — fo oily E log we) + 2[y cot”"(y/#)-H] ; 
— oo 
The expression in (13) is the potential for the flow caused 
by a line distribution of sources on the centerplane and on the above- 
water image of the centerplane, the potential having been evaluated 
on z=0. We recall that we started constructing our solution, in 
Equation (1), by assuming that there was a distribution of sources on 
the submerged part of the centerplane and a distribution of opposite 
sinks on the image of the centerplane. We now discover the interest- 
ing fact that one part of the potential, when evaluated on the plane of 
the undisturbed free surface, represents a symmetrical distribution 
of singularities, rather than an antisymmetrical distribution. The 
symmetrical distribution would have been a logical starting point in 
the ordinary slender-body theory, in which a rigid-wall free-surface 
condition must be satisfied. It appears in the present analysis asa 
natural consequence in the region downstream of the bow region, al- 
though we started with quite a different picture of the flow around the 
bow. 
b) The remaining part of the expression in (7) is oscillato- 
ry with respect to x, and so we use the procedure that worked well 
in estimating the downstream behavior of ¢, and £ . We go through 
the following steps 
fos Peers (1 - oH) cos VEL] x 
27 xe p2 
1502 
