Cases of Three-dimensional Fluid Motion. 363 
“ prominence of the interference effects depends upon the relative magnitudes 
of the constants f and i; the example we have chosen shows a pronounced 
effect due to the final maximum of interference being near the maximum of 
the mean curve. 
8. We may note briefly the interpretation in terms of a submerged body 
The surface of the body is one formed of stream-lines due to the two equal 
doublets in a uniform stream; the axes of the doublets are in the same 
horizontal line at a depth /, which must be large compared with, at least 
the vertical dimensions of the body. For instance, with suitable relations 
between the constants, the result would give the wave resistance of two 
small bodies, of nearly spherical shape, one behind the other at a distance 
large compared with their dimensions. 
9. By combining simple symmetrical pressure systems, we may generalise 
the previous results; this seems an easier process than the direct discussion 
of unsymmetrical systems. We shall assume that the component simple 
systems are all of the type (12) and have the constant f of the same value, 
and that the centres of the systems all lie on the axis Oz. 
In the first place we must extend the analysis of §8 to two components of 
unequal magnitudes A and B, with their centres at the points (A, 0) and 
(x, 0) respectively. From the argument expressed in (29)-(32), it is easily 
shown that the value (33) for the wave resistance must be altered by replacing 
A? cos? (xo sec h) by 
4 [A?+ B?+ 2AB cos {ko (A—&) sec } J. (34) 
Suppose further that the pressure system is given as a continuous line 
distribution of components along Ox in a range from / to fe, the magnitude 
of the element with its centre at (#, 0) being proportional to some function 
vv (x); in other words, suppose the surface pressure is given by 
fg (h) dh 
pe Lill ese eP 
the function y (/) being such that the transformations used in the preceding 
analysis are permissible. For the system (35) we have to sum (34) for all 
possible pairs of elements; this is performed by taking the double summation 
3 jvm dh [.¥® cos {xo (h—k) sec p} dk. (36) 
(35) 
The wave resistance for the system (32) can be completed now from (33) 
and (36); we have 
R = (4r/gp) 00°? 
| aya | oR | “F008 he- 20/00" cos {iy (h— lk) sec} dg. (37) 
hy hy 0 
155 
