Wave Resistance: Effect of Parallel Middle Body. 89 
9. It is not necessary to graph a large range of resistance curves at each 
speed to find the positions of the maxima or minima. Turning to the general 
expression (10) for the wave resistance, we require the roots of the equation 
dR/dk = 0. (29) 
Since P’,41 = - we find that this reduces to 
in poets OS 
¢ ce \(g@k+))\ ., (2gk\ _ 
ee GE trea | alm aay 
But we have 
pPs(p) = 4P3 (p)—pPs (p)+5Ps (p); (31) 
and if we write 
a = gic? = Qnl/rA;  -y = 2ghk/c? = 4rck/d = 2ka/l, (32) 
the equation (30) becomes, in terms of functions which have been tabulated 
here, 
y¥_p Te 5 Pa) om) 
( acy + 2a? : oat ~* ' Qy+4a ° tae) 
r(bp—Lp)iyto+sbPfy}=9. (8) 
The problem is the determination of pairs of positive values of 7 and y 
satisfying this equation. The approximate formulz (27) and (28) are equivalent 
to arranging these in series in linear relationships. For a numerical study of 
the roots of (33), we have to use the tables and graphs of the P-functions to 
which reference was made in $5. Starting with some value of x, we find the 
corresponding value of y from (33), and it is not difficult when we take another 
value of x to decide which is the corresponding root in y; the preliminary 
survey of the curves in fig. 2 enables us to follow out any required sequence. 
We choose here the series corresponding to » = 1—that is, the series of crests 
which includes those marked A, in fig. 2. It was found that with the large- 
scale graphs of the P-functions, the value of the left-hand side of (33) could 
be calculated with sufficient accuracy for a graphical method to give the 
required root; except that for high velocities—that is, low values of z—the 
graphs had to be supplemented by direct calculations. 
Omitting the details of the work, the following pairs of roots were obtained :— 
x 5-97 5 4 3-6 3 2-5 2-3 2 1:6 
3°83 4-7 5-36 | 5-66 6-1 6-56 
226 
