Wave Resistance : Some Cases of Unsymmetrical Forms. 235 
distribution of strength f (x) along portion of the axis of 2; then, with @ as 
distance from Oz, the velocity potential and stream function are given by 
é: F(a) a 
toot |e ap 0 
— reg? 4. [ _(@ MFO ah : 
y= yea" + | { (eA + oP, a 
The form of the solid is obtained from the equation V=0. The graphical 
method is first to graph the integral in (2) upon @ as a base for given values 
of x, obtaining a family of curves each corresponding to a constant value of 
x; then on the same diagram the parabola Y= $c@* is drawn. The inter- 
sections of the parabola with the family of curves give pairs of corresponding 
values of x and @ on the zero stream line. 
It is obvious that if 7 (4) is finite, not zero, at an end of the range of sources 
then the body has a blunt end ; and further, the length of the body is greater 
than the length of the range. If f (h) is zero at both ends, the body has a 
sharp point at both ends and its length is equal to the length of the range ; 
if, in addition, f’ (A) is zero at an end, the sharp point at that end is one of zero 
angle. 
3. In considering the second approximation, namely, the image of the 
distribution /(/) in the upper free surface of the stream, it is more convenient 
to use as the elementary system a doublet with its axis parallel to the stream. 
As we are dealing with solid bodies of finite size, we can in general replace the 
line of sources and sinks by an equivalent line of doublets ; thus instead of (1) 
we have 
pace y [ean etna 
J{G—hF =o} 
provided J’ (kh) = f(A). and wv (h) is zero at both limits. Consider now a solid 
of revolution with its axis horizontal and at a depth f below the surface, the 
form being such that the image of the uniform stream in it is a line of doublets 
(3) 
of moment | (i). The image of this system in the free surface can be shown 
to be a certain distribution of doublets of infinite extent along a line at a height 
jf above the surface. For the present purpose we shall quote the expression for 
the wave resistance* 
r i 
R = 16agiee* [y () dh | Yh) ah’ | see® g 
Jo : 
X eos [{g (kh —h’)/c*} see d]e- CHO Fdd. (4) 
* <Roy. Soe. Proe.,? A, vol. 95, p. 363 (1919). 
242 
