498 Dr. T. H. Havelock. 
On simplification this leads to 
X = 2me~* (bcos ea—a sin xa)/(a? +0). 
Hence from (7) the corresponding resistance is given by 
(0+ 0?) ypR = 41?x?e~« sin? (xa—e), (30) 
with tane = b/a. 
We have in (30) a form very similar to those we have already studied. 
The phase € means a bodily displacement of the series of humps and hollows ; 
but, again, ¢ is small under the usual circumstances, when the difference 
between yw and d is small compared with either. 
Further, because of the symmetry of the distribution fore and aft, there are 
values of «, with corresponding speeds, for which R is zero; we have seen 
that to avoid this result we must suppose the magnitude of the pressure p to 
be less in the vicinity of the run than at the entrance. We could introduce 
this want of symmetry by considering 
an &—¢? ( 
se (Cea (Cer ea 
In the expression for the integral y we should have a part corresponding 
to each of the poles a, +70, and d2+1b2; in consequence, the resistance R 
would be similar in form to the expression in (23). 
From (30) we notice that the wave-lengths corresponding to humps on the 
resistance curve are submultiples of 2a; also when 2X and » are nearly equal, 
2a is of the order 2, the distance between the two positive pressure humps. 
The typical interference effects in this example are due to the interference of 
the bow and stern systems; in order to get a secondary interference effect 
between the positive and negative parts at the bow these must have separate 
individuality to a greater degree, as we saw in §5. For instance, we could 
consider two distributions like (20), one associated with the entrance, the 
other reversed and associated with the run; we should then have a very 
general type of distribution represented by 
31) 
oe Eee A (EERE a (32) 
(E—2Y)— Bi (E—31) + 4ay* (E441) + Bo (E441) +43” 
It is unnecessary to graph this or to put down expressions for y and R. 
We should obtain a sum of expressions like (23) involving sines and cosines 
of 2«a, and of 2«a2, and, in addition, of «il. There would be in general 
various possibilities of subsidiary interference effects; the main one would be 
the bow and stern interference represented by «/, and the next in importance 
that between the positive and negative parts at the bow represented by 2 «a. 
There would also be the possibility of these two effects adding together at 
certain speeds. 
103 
