92 
PROFESSOR W. M. HICKS OK VORTEX MOTIOX. 
is done at once on the squared paper). Mark the points where the radii vectores to 
these points cut the corresponding circles. Join these points by a continuous curve, 
and the shape of the particular \p curve is obtained ; call it the xjj^ sheet. This first 
curve should be obtained with care and as much accuracy as possible. We may now 
proceed to draw from this as many of the other sheets as we please. Suppose we 
want to draw the curve xjj = k. xjjo. We set a pair of proportional compasses (or any 
similar method) to the ratio \/10k. Suppose the xjji cuts any particular circle at P, 
set the short legs of the compasses to its abscissa. Turn it round and find the point 
on the same circle whose abscissa is the new value. Proceed thus with the other 
circles and the sheet is rapidly traced. Although this may appear cumbrous in 
stating, it is very expeditious in practice, and with a moderate amount of care very 
accurate. 
Having traced the \fj curves, we may now easily trace the proj ections of the stream 
lines, for these are given by 
= fof-. = (see fig. 4), 
Fig. 4. 
26. It will be interesting to go into further details for a few cases, and for this 
purpose we take the first two aggregates of the X2 and K families. 
The distinguishing feature of tlie Xo types, is that the aggregates are at rest in the 
surrounding fluid. The distinguishing feature of the Xj types is that the vortex and 
stream lines are coincident. 
Xo aggregates. Here 
U= 0, 
M = 
mjjb — A, sin A 
15 ^i\ — sin A 
E C) o 
= T5 « 
A- sin- A 
(S'iA — sin A)- 
A- sin- A 
A - sin Xf 
where Eq is the energy of a Hill’s -^.^p egate of equal volume and circulation. 
