84 T. H. Havelock. 
In this case the arguments of the P functions increase by intervals of 0-8 
from zero up to 10. 
It may be noticed from (17) and (19) how the relative importance of the 
oscillating terms alters with the velocity. 
This process was carried out for nine different velocities, namely :— 
gle? = 0-1, 0-0625, 0:05, 0-045, 0-04125, 0-0375, 
0-03125, 0-02, 0-0075. (20) 
Five of the curves are shown in fig. 2, which gives the quantity 8cR/gpb7 
on a base 2 k representing the length of parallel middle body ; the curves for 
higher speeds are not reproduced, as the scale would obscure the effects, but the 
data are used in the discussion. 
Approximate Formule. 
7. It is convenient to summarise now the experimental data and empirical 
formule derived from them. 
The investigation of W. Froude* was the first direct study of interference 
of bow and stern waves made by testing models with the same bow and stern, 
but with increasing lengths of parallel middle body. We associate with this work 
the subsequent paper by R. E. Froude,} who applied the principle of interference 
to the resistance of a given model at different speeds. Founded on this work, 
the approximate theory has been developed : the bow produces a wave system 
beginning, so far as regular transverse waves are concerned, with a crest 
slightly aft of the bow, while the stern originates a system beginning with a 
trough a little aft of the after-body shoulder. Assume that this wave-making 
length, say Z, is approximately independent of speed, and further assume that 
the wave resistance is chiefly due to the transverse waves. If, then, A is the 
wave-length of regular transverse waves for velocity c, the so-called humps 
and hollows on the resistance curve occur at speeds for which Z is an odd or 
even multiple of $2. Or, if we assume an approximate formula 
R = A—B.cos (gZ/c*), (21) 
where A and B are undetermined functions of velocity, the humps and hollows 
correspond to the maxima and minima of the cosine factor ; hence we have 
the sequence 
1 1 1 
1, — — aa 22 
* W. Froude, ‘Trans. Nav. Arch.,’ vol. 18, p. 77 (1877). 
7 R. E. Froude, ‘ Trans. Nav. Arch.,’ vol. 22, p. 220 (1881). 
