Wave Resistance: Effect of Parallel Middle Body. 83 
of the three functions over the range from 0 to 40, the scale for p being 1 inch 
for unity and the scale for the ordinates being 10 inches for unity ; these gave 
the required accuracy, supplemented at critical pots by numerical calcula- 
tion. The graphs are not reproduced here, as they lose their practical value 
unless on a very large scale ; they are, of course, similar in character to graphs 
of the Bessel functions—oscillating curves diminishing in absolute value with 
increasing argument. 
Resistance Curves. 
6. We can now make a numerical study of the wave resistance given by 
(10). We might adopt dimensionless variables, such as gl/c? and k/l, but the 
calculations were begun with the intention of comparing the results with 
W. Froude’s curves ; we take therefore 
1 = length of entrance = length of run = 80 ot (15) 
2 k = length of parallel middle body, 
with 2k increased from zero up to 340 feet. For an assigned velocity c, the 
values of R were found for every 20 feet of parallel middle body ; as a rule, 
intermediate values were also calculated so as to define the maxima and minima 
with sufficient accuracy. 
Two examples of the work may suffice. With 
gic? = 0045; c= 26-75 ft./sec.; V = 15-83 knots; (16) 
we have, from (10), 
als 
3°6 
sails 
25-92 
eubt) 
25-92 
2 
R = 921 10-974 x [0-5026-+ (Ps— gg Pit se", Ps}{0-00k-+7-2} 
TT 
i 1 ) | 
—— P, — ——— P| {0-09%-£3- : . 
teal: mae +3-6}+ P; {0-094} |. (17) 
The notation { } denotes the argument of the P functions in the preceding 
bracket. 
For increments of 10 in the value of k, the P functions were required at 
intervals of 0-9 from zero up to 22:5. Again, with 
gic = 0-02; ¢ = 40-18 ft./sec.; V = 23-76 knots, (18) 
we have 
ee 1 
— 9964 19 [o-2344 (QP— = a 
Hoa roRm ACK TG eg 5-12 
Ps) (0-04b-+3-2} 
1 
5-12 
1 1 
apes 
1-6 ° 2-56 
an ( Ps) {0-04k-+1-6} + Ps {0-04h} |. (19) 
220 
