934 T. H. Havelock. 
The remaining sections deal with the similar problem for a model of infinite 
draught and constant horizontal cross-section; the forms of the section for 
the two cases are shown in fig. 1. Here, with the help of tables and graphs 
available from previous studies, the expressions for the wave resistance have been 
graphed and the curves are shown in fig. 2. The result of smoothing the lines 
of the rear portion is very marked. It makes the curve like experimental 
ones in this respect at least, that the curve is a continually ascending one in 
the range shown ; the superposed oscillations are not large enough to make actual 
maxima and minima. A more complete study of the progressive effect of 
small changes in the rear half of the model would involve very lengthy calcu- 
lations ; the examples given have been chosen for the comparatively simple 
form of the mathematical expressions. It is to be understood that they are 
not intended as a direct representation of the actual effects of fluid friction ; 
but they show the great difference in interference effects which are produced 
by an asymmetry of the general nature suggested by them. 
2. The fluid motion produced by a body entirely submerged in a uniform 
stream may be investigated by the method of successive images. The first 
approximation consists of the distribution of sources and sinks which is the 
image of the uniform stream in the surface of the body; the second is the 
image of these sources and sinks in the upper free surface of the stream, and the 
process could be carried on by successive images in the surface of the body and 
the free surface of the stream. After the second stage the expressions become 
very complicated, as the image of a single source in the upper free surface is a 
distribution of infinite extent along a horizontal line at a height above the free 
surface equal to the depth of the source. It would be of interest to carry the 
process further in some simple cases, but at present the second stage must 
suffice ; it can be seen that, in general, this implies that the ratio of the maxi- 
mum vertical diameter of the body to its depth below the surface must be 
small. 
For the first stage of the approximation, instead of finding the image system 
for a given form in a uniform stream, it is more convenient to begin with a 
given distribution of sources and sinks and deduce the form of the body. As 
we shall deal only with surfaces of revolution, we assume a line distribution of 
finite extent along a line parallel to the stream. Writing down Stokes’ 
current function, the form of the body may be found by graphical methods 
devised by Rankine and applied to shiplike forms by D. W. Taylor and other 
writers. 
Let the stream, of velocity c, be parallel to Oz, and let there be a source 
241 
