465 T. H. Havelock 
5» will give no effect over o, if o» is aft of o,, but the full interference effect 
of the two systems will be added to R, if o, is aft of o5. 
Summing up this general discussion, we see that the total resistance 
of each system consists of various parts: the resistance of each as if exist- 
ing alone, mutual actions between the two systems which are equal and 
opposite and may be classed as due to local disturbances, and wave inter- 
ference acting on that system which is to the rear of the other. It may 
be noted again that in this analysis we are assuming the source distri- 
butions to be given. It has been shown how the various terms in the 
Tesistances can be calculated when the two systems are in one and the same 
vertical plane. A similar analysis could be made for more general cases; 
but we shall consider in some detail a simple distribution consisting of two 
isolated doublets. 
6—Suppose that there are two equal horizontal doublets A, B each of 
moment M in the liquid at the points (0, 0, —f) and (—/, 0, —f) respec: 
tively; thus A is directly in advance of B. If the points are sufficiently 
far apart, the corresponding bodies would be, approximately, spheres each 
of radius b given by M = 458c. However, all we shall assume meantime 
is that the doublets are far enough apart to represent two distinct bodies, 
one enclosing each doublet, whatever their actual shapes may be. 
The velocity potential is given by 
$= cx + ox + dp, (14) 
where 
Mx Mx , ik ,M (7 0 g—K(f—2) +ix (2 cos 6-+y sin 6) 
= SS ee CS) AR) |W eee Se : 
He ri? r° Ps Te [/_see \, K — kp sec? 0 + ip sec 0 ig 
(15) 
and ¢, is a similar expression with x + / instead of x, the notation being 
the same as in (1). 
The form which (3) takes for an isolated doublet is 
od 
R = — 479M Bye? (16) 
where in 0?4/0x? we must omit the term in ¢ due to the doublet at the 
point in question. Thus we may calculate the resistances R, and R, 
separately. In the process we have to evaluate the expression 
li as 6 dé [oe) en 2aftine cos 6 5 A (17) 
—limi| cos 8 fe. 
u—0 [L [[ kK — Ky sec? 6 + iu sec 0 
413 
