467 T. H. Havelock 
to the rear system. In addition we have the terms R’ and R’’, which may 
be regarded as a local action and reaction, their magnitudes diminishing 
rapidly with increasing distance. It may be noted that with M pro- 
portional to the velocity c, R’ increases as the square of the velocity; this 
may be associated with the fact that, although the regular wave system 
diminishes to zero ultimately with increasing velocity, there is a permanent 
local surface elevation. 
7—Suppose now that the two doublets are abreast of each other at a 
distance 2k apart, that is, suppose equal doublets A and B at the points 
(0, 0, —f) and (0, 2k, —f) respectively. The velocity potential is 
p= cx + oy + op, (25) 
with ¢, given by (15), and ¢, by a similar expression with y — 2k instead 
of y. 
We have 
SS. Ob, , Obs 
R, = — 4npM ( a + oo ), (26) 
evaluated at the point A and omitting from ¢, the term representing the 
doublet at A. 
It is clear from the symmetry of the arrangement, that the local terms 
give no effect; reducing the remaining terms we obtain the result 
R, = Ry + 167 9K 94M? [- sec’ Bee"? Cos (2kok sin 8 sec? 6)d6, (27) 
0 
with Ry given by (20). 
Taking M = 4b’c, we may regard this as the resistance of a small 
sphere at depth f in a stream and at a distance k from a vertical wall 
parallel to the stream; it is of some interest to estimate the influence 
of the wall upon the resistance. Ry has been expressed previously in 
terms of Bessel functions; it is given by (using the notation of Watson’s 
Treatise on Bessel Functions) 
= it || 1 } 
Ry = Ee |Ko(#) + (1 + 5-) Ki}, (28) 
with « = kof = gf/c. 
The integral in (27) is equal to 7. 07X/da?, where, with B=x,h, 
mw /2 
= | sec 0e—2«°* 8 cos (26 sin 8 sec? 6) dé 
.v) 
= | © e-20.c0sh*« cos (8 sinh 2u) du = 4e~K, (Ve + BD. (29) 
0 
415 
