Cases of Three-dimensional Fluid Motion. 361 
The terms Rjz and Ra represent the interference effects. Let B, and Bo 
be the centres of the two systems p, and py». To calculate Ra, consider a 
constituent plane wave-front through B,; take this line as a new axis O’7’, 
and a perpendicular line through Bz for the axis O’z’, as in fig. 1. 
Then, corresponding to the expression (15) in § 4, we have as an element of 
Re the quantity 
ut eu cea ic A cos (xox sec? p) div’ 
2 3 Ko fsec? If \ 
Aor ky" A sec? he i \w ( {(@’ + 2h cos $F +924 f) (30) 
The similar element in the value of Riz is the expression (30) when we 
have replaced 2’+2hcos@ by «’—2hcos¢. Adding the two elements and 
carrying out the integration with respect to y’, we have, as an element of 
Ri2+ Ray, 
2A2 acs gy e—Kof sec? * COs (Heoer" sec* ) dan’ 
Sm fKo A? sec? pe Hos @ + 2heos pF 472 
= 87°? A* sec? He- nose" cos (2eoh sec). (31) 
Replacing from (11) the proper factor and integrating with respect to ¢, 
we have 
ar /2 
Riyo+ Rig = (87/ 9p) xo*A? | sec? pe-*Fse*4 cos (QKoh' sec) dd. (32) 
0 
Finally, from (29) and (32), the total wave resistance R is given by 
7/2 
R = (167/gp) Ata | sec? he-**oFse*d Gos? (Kyh sec p) dd. (33) 
0 
We can express R in series of known functions by expanding cos(2x«oh sec d) 
either in powers of xo, or in Bessel functions J, (21); however, as these 
series involve either Wi,m(2«9/) or K,(«of))t they are of no use for numerical 
calculations. 
It is not difficult to calculate numerical values directly from the integral 
(33) for given values of the constants. To obtain a graph showing the 
T See note by Editor on page 150. 
153 
