364 Prof. T. H. Havelock. 
10. Consider as an example the case when w(/) is constant, so that the 
surface pressure 1S 
y dh 
plAf = l. {@—hp+pPtfey? 
ee x—h A, eth 
(PAP) @-AP+P TPP? (PAP) E+hPE P+ Py? 
This may be regarded as the combination of two equal systems of opposite 
sign, with their centres at the points (A, 0) and (—h, 0) but not symmetrical 
round these points. 
In this case, after carrying out the integrations with respect to h and & 
(37) gives 
(38) 
ar [2 
R = (16 7/gp) A? ko { sec? dh e~?noSsec?$ gin? (x9 sec h) dd. (39) 
0 
The integral may be treated similarly to (33). One of the main differences 
lies in the factor sin? (xoi sec p) instead of cos? (xo sec pd); this is because 
we have now two equal positive and negative systems instead of two positive 
systems, and in consequence the series of humps and hollows on the resistance 
curve will be interchanged. 
We have chosen this case partly because of the corresponding problem in 
the motion of a submerged body at depth 7. Integrating a line of doublets 
of constant strength results in a simple source at one end of the line and an 
equal sink at the other. Hence, the submerged body is one of the oval- 
shaped surfaces of revolution formed by combining a source and sink with a 
uniform stream; it follows that, as in §5, the strength of the source is 
Ac/2gp. It may be noted that the coefficient A in (39) has different 
dimensions from that in (33), agreeing with its introduction in (38). By 
making / small in (39) we regain the former result for a sphere. 
11. If a prolate spheroid of semi-axis a and eccentricity e is moving in an 
infinite liquid with velocity ¢ in the direction of its axis of symmetry, it can 
be shown that the velocity potential may be written in the form 
ae (a?e?—h?) (a—h) dh 
b= Al” Gaara ey 
where A = 1/[4e/(1—e?)—2 log {(1+e)/(1—e)}], and where we have, for the 
moment, taken Ow along the axis of symmetry of the spheroid. This 
expresses ¢ as due to a line of doublets ranged along the axis between the two 
foci. Hence the surface pressure corresponding to the motion of the spheroid 
with its axis at depth 7 is 
(40) 
ze (ae? —h?) dh 
p= 20th | eairersr Te “) 
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