469 T. H. Havelock 
Carrying out the calculation as before, we find 
peaicteeee| 
(2  f2)9!2 J 
m2 sin 6 d0 i Kg sec? 0 sin Bliss li cos 2mf 
0 m + "sect 0 
Y = 7M — 
e72mk sin @ m3 dm 
— 16x yeM? | 
0 
m2, . s 2 : 
= 16ror'Me | sin 0 sec® Oe-7*-Fsee*? Cos (2kok sin 0 sec? 0) dO. (32) 
0 
Here again with M = 4b%c, the first term is the usual approximation for 
the attraction between two spheres moving abreast in an infinite liquid 
at a distance 2k apart. 
9—When one sphere is directly behind the other, the oscillating part of 
its resistance is due to the transverse waves in the pattern made by the 
leading sphere. When the two spheres are abreast of each other, there 
are no similar oscillating terms. We shall now consider the more general 
case of any relative positions, when in suitable circumstances we can 
distinguish between interference due to transverse waves and diverging 
waves. 
With the same notation, we take the doublets A and B to be at the 
points (0, 0, —f) and (—J, k, —f) respectively ; thus, with / and k positive, 
B is a distance / to the rear of A and a distance k to one side. The 
velocity potential is 
p= cx + bat dp; (33) 
with ¢, given by (15), and ¢, a similar expression with x + / instead of 
x and y — k instead of y. Each resistance is given by the expression in 
(26), evaluated at A or B in the manner already explained, and the calcu- 
lation of the various terms follows the same lines. 
For the term corresponding to R’ in (21) we now obtain 
f APS Ne 21? — 3k? — 12f?) (34) 
UP ane key (P+ R+ 4fp2yi2s 
The remaining terms are more complicated than in the previous simpler 
cases; for their contribution to Ry we have to evaluate an expression 
R’ = 127eM?/ 
.(™ co ex (Leos 6—k sin 6) 
i | cos 8 d0| Bl RE Cutie PS NE es eS hies (35) 
a » K — Ky sec? 8 + iy sec 0 
We first reduce the integration in 0 to the range 0 to 4x. Then the 
various integrals in x are transformed by contour integration, the form of 
the results depending upon the sign of /cos 8 — k sin 9; this involves 
417 
