Vortex Theory for Bodtes Moving tn Water 
not equal but are chosen so as to make the distance travelled by the 
nth shed vortex equal to the separation of the first vortex from the 
edge, but there is still some zigzagging. The real problem of this 
kind of discretisation is that when the discrete vortices are spaced too 
closely, when they get close to each other, they indulge in a planetary 
motion about each other ; and if they are spaced too far apart you will 
not get the velocity field right and the shape of the sheet will not deve- 
lop properly. 
Figure 4 shows our current recipe for this sort of situation, 
and that is that after each step of the calculation one moves the vor- 
tices to approximately equidistant positions while maintaining the 
centre of vorticity of the whole lot in the correct position. 
Figure 5 shows a comparison with the only exact solution 
known to myself, von Wedemayer's. The circles on that graph, using 
the equispacing recipe, seem to give excellent agreement for the de- 
velopment of circulation with time for the vorticity shed from one of 
the edges. The total circulation shed into the stream is, of course, 
zero, and that is one of the big troubles in this game. 
Figure 6 shows what can be done with this method. Here a 
plate has completed one half cycle of oscillatory motion normal to it- 
self and Figure 7 shows the disposition of the vorticity after a com- 
plete cycle. I will not bother you with estimates of force associated 
with this kind of thing or conformal transformations, but I would like 
to pass to the last example. 
This shows the case ofa plate, Figure 8, a bit like the earlier 
one, growing in length vertically while moving normal to itself. We 
have applied that solution to slender lifting surfaces of arbitrary plat- 
form (arbitrary, that is, while still remaining slender) and camber and 
we seem to get rather better results for that type of lift surfaces with 
leading edge vortex separation than the results obtained over the years, 
first by Legendre here and then by Brown and Michael and others. So 
we are not completely put off by Birkhoff's warning and I would recom- 
mend our particular recipe for calculating the development of vortex 
sheets, although we have not, naturally, got anything on uniqueness 
for the results. 
We are now working on a number of configurations of interest 
where there are sharp edges to fix the separation, such as that shown 
in the first picture, and also on others, where non-linear lift pheno- 
mena occur. 
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