533 T. H. Havelock 
Hence from (32) we have 
R= 407 9px a4 f(k)f* (ky) e- * (34) 
With 7 
SF (Ko) =b,+ bh (oa) =P a (koa)? + ..., 
and with the equations (20), it could presumably be shown that (34) is 
the same as the real part, with sign changed, of the expression (28). 
However, it has been used here simply to verify the previous expansion; 
substituting from (23) and (24) we obtain from (34) the same result as is 
given in (29). 
6—We may now examine the expressions (29) and (30) numerically. 
It is easily seen that if the ratio a/f is small, the first term in each case 
gives a close approximation at all velocities. Further, the ratio of the 
second term to the first in (29) and in (30) is —2r,«,a?, that is 
tat 
with « = 2x, f = 2g¢f/c?. 
The quantity in brackets in (35) approaches the value —1 as c becomes 
zero and the value +1 as c becomes infinite. It has a maximum negative 
value of —2-:57 at «= 4-5 approximately, and a maximum positive 
value of 1:9 at about « = 0-6. Hence the effect of the second approxi- 
mation in (29) is to increase the wave resistance at low velocities and to 
give a rather smaller value at high speeds. 
Taking a/f = 4, as a moderate value of this ratio, and calculating the 
resistance from (29), it is found that the value does not differ by more 
than about 9% of the value of the first approximation at any velocity. 
As an example of the numerical values in this case, for « = 6, that is for 
c= 0:58 \/(gf), the following are the values of the successive terms in 
the expansion in square brackets in (29): 
1 + 20 — 2a%e~- li (e*)}, (35) 
1 + 0-0746 + 0-0134 + 0-0015 + 0-0001. 
Another case which has been worked out in some detail is a/f = 4, this 
being definitely outside the range of the first approximation for the most 
part. Numerical values were calculated for both X and Y for « = 8, 6, 
5, 4, 3, 2:5, 2, and 1. On account of slower convergence of the series at 
the higher values of «, an estimate was made of the next term beyond 
those shown in (29) and (30). The results are shown in fig. 1. 
The curves R and Y are the wave resistance and vertical force calculated 
from (29) and (30); R,, Y; are the curves given by the first approximations, 
427 
