284 T. H. Havelock. 
A similar method was used for the integrals in (37) and (38). For (39), 
the Bessel function was expanded in powers of (1 + ¢?), and integration carried 
out term-by-term ; the integrals involved are then of the form 
2n+1 
i (1 + #) 2 e-?" dt, (44) 
which can be expressed in terms of the Bessel function K,, defined in (41). 
By recurrence formule, the terms can be reduced to expressions involving 
K, and K,, and tables of these functions are available. In all these calcula- 
tions no attempt was made to obtain any high degree of numerical accuracy ; 
the object was to obtain sufficient values to enable graphs to be drawn showing 
the nature of the results and the main differences between the two series. 
The graphs are shown in figs. 2 and 3; the scale is the same throughout, the 
ordinates being R/rgeb?, and the abscisse u/1/(gf). 
The nature of the results is obvious from the graphs. Fig. 2 shows the 
curves for the end-on motion. The curve B, which is the same in both 
diagrams, is for the sphere and shows the maximum just before the velocity 
(gf ¥?. The graphs for C, D, E show how much the resistance is diminished 
at the lower velocities by increasing length in this way ; but this is followed 
by a rapid increase at higher velocities. The latter effect may be described, 
roughly, as due to the final interference between bow and stern system giving 
a prominent hump on the resistance curve ; the interference effects at lower 
321 
