574 T. H. Havelock. 
satisfies all the above conditions except (3); and further that (3) is satisfied 
by the complete expression on account of the expansion 
fed= 4({[{ reo 
07070 J—-@ 
cos (1z—e) cos (n€—e) cos m (E—x) dE dE dm dn 
a | | | f (E, ©) m2e~ em" 2+519 cos m (E—2) dé dt dm. (5) 
An expansion which may be verified without difficulty, e having the value 
given in (4); it is assumed that the function /’(@, z) is such that the various 
integrals are convergent. 
The expression (4) holds for y positive. The first and third integrals 
represent local symmetrical disturbances, while the second integral represents 
the waves which follow the ship if we imagine it to be advancing into still 
water. 
If dp is the increase of fluid pressure due to the disturbance ¢, the wave 
resistance is given by 
R=—2|| 6p. dds = Bpe [| SOS aed (6) 
the integration extending over the vertical median plane of the ship. 
The first and third terms in (4) contribute nothing to R, and we have 
us Apc! oo cD p0O ce} m2e- m2 (z+.¢)/q 
R= 7g e fr alaloe tS (2) Ff & &) (nbc! ]/ ge —1y? 
cos m (w«—&) da dz dé d& dm 
_. 4iae- al P42) mdm 
TY 9 Sole Ge /(ge=1) 
Co) 
ait = | | J’ (@, 2) e- 2/9 cos max da dz 
0 J—x 
— | | SF (a, 2) e7-""7/9 sin ma da dz. (7) 
0% —w 
This is Michell’s expression for the wave resistance. We shall take the 
origin at the midship section and assume the ship to be symmetrical fore and 
aft; in these circumstances, I = 0. 
Submerged Spheroid. 
3. The application of (7) is limifed by the assumption involved in (3), that 
the inclination of the surface of the ship to the median plane y = 0 is always 
small. To illustrate this limitation we may consider a particular case in 
202 
