PROFESSOR W. M. HICKS OH VORTEX MOTION. 
91 
them the Q points. They are easily formed by fixing a lath at 0 and bending it to 
touch successive loops of the J curve. It is easy to do this correct to two decimal 
places, when numerical calculation will carry it to any degree of approximation desired. 
The points where the J curve cuts the axis of x correspond to the Xi parameters. 
We will call them the II points. 
Denote the points where the parabola through P cuts the J curve again by the 
letters p. These points give the sizes of the shells into which the aggregate divides. 
ON 
If ON be the abscissa of any such point, Xr = ON, and r = . a gives the radius 
of the corresponding interface between two shells. It is evident at once from the 
construction that the thicknesses of the shells, as we pass in or out, are alternately 
greater and less—that there are two categories, in one of which the thickness 
increases as we pass in, and an alternate series in which it decreases. There will be, 
however, some irregularity in the two inner components. 
The position of the equatorial axes is determined by those abscissa, for which the 
tangents to the J curve and the parabola are parallel. They are easily recognized by 
the eye, and thus a starting point for calculation is readily obtained. The difference 
of ordinates of these points (P1P2) is proportional to the secondary circulations of 
the corresponding shells. In fact, when multiplied by 7^p,/(S^X — sinX), the products 
give the values of those constants. It is therefore clear from the figure that these 
circulations are in opposite directions alternately, and that we get two alternate 
series of ascending- and descending values. 
The function S (X) = SiX — sin X denotes the area between the J curve and the 
axis of X up to the point X. It is clear, therefore, that it has its maximum values at 
the odd Xj points, and its minimum at the even ones. 
The tracing of the current sheets is particularly easy from the fact that they are 
given by functions of the form 
xjj = Y (r ). slir 9. 
Let 
and let \po and Vq denote values at the equatorial axis {i.e., xJjq a numerical maximum). 
Then 
^ _ / (^’) 
^0 ~/bo) 
siiP 9, 
t_ /(To) 
^0 ’ / (’’)' 
On squared paper, draw a series of circles, radii sub-multiples of a, say at intervals 
of '05a or "la, also the circle r = This last circle has the property that all the 
current sheets cut it at right angles. 
Let us trace first one sheet (say xfj = 'Ixfjo). We do this by tabulating the values 
of sin 9 for values of r, corresponding to the series of circles drawn. Now mark on 
the bounding circle (r = a) points whose abscisste aie those tabulated values (which 
N 2 
