10 T. H. Havelock. 
where F is the function specified by (28) for positive and negative values of the 
argument. 
We may now complete the expressions to include a distribution in which 
there are ranges of continuous change of gradient. It is obvious from the 
preceding argument that the complete expression is 
ca Su { |S 
where the summation covers all points of sudden change of slope and all ranges 
aM 
“F (@ — hes) + \aa F(c—h) an, (33) 
of continuous variation. 
The function F can easily be tabulated and graphed by means of Q,) and Pp. 
In summing and integrating in (33) it is to be noticed that the Q, terms are 
symmetrical before and behind each element, while Py only exists in the rear 
of each element. When the distribution M is a sum of integral powers of h, 
it appears that (33) can be expressed in terms of the P functions defined in 
(26), for the wave disturbance, together with a similar series of Q functions 
for the local disturbance. But even if M is not given in simple analytical 
form, the elevation could be calculated directly from (33) by numerical or 
graphical methods of integration. 
7. We have been discussing the fluid motion due to a given distribution of 
doublets, the surface elevation we have calculated being one of the stream lines. 
It would be of interest to trace, if possible, other stream lines so as to exhibit 
the form of a submerged solid to which the given distribution is equivalent ; 
but the calculations would be lengthy, even in the simplest cases we have 
considered in the previous sections. For a ship model we have already men- 
tioned the usual approximation for the equivalent distribution of doublets 
when the ratio of beam to length is small enough. For a model of infinite 
draught, whose horizontal half-section is given by y =f(h), we have 
Gel (34) 
Hence (33) gives 
ca2zlirmi re-tot+|s Fema}. G9 
We note that the magnitude of the contribution due to an angular pomt on 
the model is directly proportional to the change of slope that occurs there. 
8. We may illustrate the general result by considering briefly a model with 
