The Effect of Shallow Water on Wave Resistance. 503 
adopted. With p =h/l, and « = xh, using the relation between « and @, (14) 
can be put in the form 
AnA?al? p72 gil2¢~2e/P coth V? « 
R= gplip*? ( a? +e coth a —«? coth? « dp, (17) 
with a coth a = px sec? ¢. (18) 
For a given value of p, the integrand of (17), which we may denote by /(), 
was calculated for values of « ranging from zero to 3 at intervals of 0:2, and 
in certain cases also at unit intervals up to the value 10. Taking next an 
assigned value of x, the value of ¢ corresponding to each value of « was found 
from (18). The integrand f(a) was then graphed on a base of @, giving a 
curve for each value of «; the area of the curve was taken by an Amsler 
radial planimeter, and then the value of (17) was obtained. The calculations 
are rather lengthy and it is unnecessary to repeat them here. 
The process was carried out for p = 2, 1-43, 1, 0:75, with about a dozen 
values of 2 in each case; some estimates were also made for p= 05, to 
confirm the general deductions. Further, the values for p= co were 
calculated from (16). The results are shown in the figure, where the unit for 
R is 47rA?/gpl%, and for ¢ is ,/(g/). 
0-4 0-6 0-8 ie) 2 4 
5. The curve for deep water, p = oo, has a single maximum at a velocity 
slightly less than ,/(gl). At this velocity the corresponding length of 
196 
