470 Prof. T. H. Havelock on the 
the term pdd¢/dt in (2). Similarly, let P be the additional moment of 
this pressure about the axis through G in the direction of @ increasing. 
Then we have 
B=—gph | e cos (ct--Kx)ndS,, . . . .. - (8) 
P=gph i | e cos (ot-++Ka){l(z—c)—nz}dS). . . (9) 
Hence we may write (7) as 
K,0B «,0P 
The usual] approximate equations for the motion of the ship are obtained 
by taking into account also the hydrostatic buoyancy and moment arising 
from the term gpz in (3). With M, I as effective mass and moment of 
inertia, respectively, and assuming a simple law of damping in each case, 
the equations are 
MGS EIN (CeGjaNG=sIB, gg og 2g yo (itil) 
NDT OD 6 sk es 4 3) OP) 
A being the area of the water plane section and m the metacentric height 
for pitching. 
When calculated from (8) and (9), B and P are of the form 
B=B, sin (cf-+«); P=P, sin (ct+a’), . . . . (18) 
Bo, Po, «, «’ depending upon the wave-length and the form of the ship. 
To obtain from (10) quadratic terms giving a mean value different 
from zero, we need consider only the forced oscillations in £ and 6. These 
are given by 
€=kB, sin (ot+a—f) (14) 
0=k’P, sin Ba tas 
k, k’ being the usual magnification factors, and f, B’ the corresponding 
phase lags, obtained by solving (11) and (12) for the forced oscillations. 
Using (13) and (14) in (10) and taking mean values of the quadratic 
terms, we obtain for the mean backward force 
R=4kkB,?sin B+4ck'P,?sin B’, . . . . . (15) 
an expression which is essentially positive. 
With ¢, and 6 the amplitude of the forced heaving and pitching, 
respectively, and By, P, the amplitudes of the buoyancy and pitching 
moment as in (13), we have from (15) 
R=4xB,, sin B+4xP)% snp’. . . . . . (16) 
4. We have only used equations (11) and (12) to show that the average 
steady force is a resistance. In attempting comparison with experi- 
mental results one cannot rely upon calculations from these equations, 
