572 T. H. Havelock. 
interesting paper, gave a general expression for wave resistance; but it 
suffers from a serious limitation, in that the surface of the ship must be 
everywhere inclined at only a small angle to its vertical median plane. 
In § 2 a short synopsis of Michell’s theory is given. 
In §3 this is applied to the case of a submerged body and the result 
compared with the work to which reference has been made; the two methods 
are quite different and have different limitations, but it appears that the 
results agree when these conditions overlap and are common to both. 
The main problem is treated by an application of Michell’s analysis in 
circumstances in which its limitations are not of serious importance, namely, 
when the body is a vertical post of infinite depth and of small beam compared 
with its length. We may regard this as a ship in which the effect of the 
vertical sides will be exaggerated, and we may study the changes produced in 
the resistance curves by varying the form of the level lines. The practical 
problems which have been kept in view in devising special cases are such as 
the effect of straight or hollow lines at the bow, the effect of finer entrance 
and increased beam while displacement remains constant, and similar 
questions. 
In §4 a set of parabolic curves for the level lines is specified so as to 
illustrate these points, and the corresponding value of the wave resistance 
obtained in general form as a function of the velocity. Certain new types of 
integral which occur in the analysis are examined in §5; they can be 
expressed in terms of the second Bessel functions Yo and Y, together with the 
integral of Yo, and are evaluated numerically by means of recent tables of 
Struve’s functions. 
In §§ 6-10, four types of model are examined, and the wave resistance 
calculated for various velocities in each case. The chief results are shown in 
the resistance. curves of fig. 2. For comparison with experimental curves 
from ship models, the base is the quantity V/,/L, where V is the speed in 
knots and L the length in feet. The models with finer entrance, or with 
hollow lines, have smaller resistance up to V/,/L=1-1 or 1:2; but above 
this speed the models with fuller ends have the less resistance. These, and 
other results of some interest agree with deductions from the corresponding 
practical study of ship resistance ; in § 11 a summary of these deductions is 
given and a comparison is made with the results of the present calculations. 
General Analysis. 
2. Take Ox, Oy in the undisturbed surface of the water and Oz vertically 
downwards; and suppose the ship to be symmetrical with respect to the 
plane y= 0. Assuming the ship to be at rest, and the water at a great 
200 
